Origami & Folding
what one sheet can do
Who found it, and when

The first thing about layers

A folder is taught four conditions at a vertex, or is taught nothing at all, and neither one says anything about the layers — which is where most of what goes wrong actually goes wrong. There has never been a rule about layer order simple enough to teach, because the question is global and every answer to it was a search. A chain of panels whose arrows all point the same way is the first one that fits on a finger.

Assumes Taught with a wrong reason and The loop a vertex cannot close.

19 min read 8 figures The pattern is the objectLocal rules, global behaviour

There are two things a person can be told about whether a crease pattern folds, and they sit at opposite ends of a very wide gap.

The first is local: at each interior vertex the angles must alternate to zero, the mountains and valleys must differ by two, the smallest sector’s two creases must differ, and the paper must be flat. Four checks per point, each of them arithmetic, all four teachable in a few minutes to anybody who will hold still for it.

The second is global: given that every vertex passes, can the panels be stacked without the paper passing through itself? That question is NP-hard, its only complete answer here is an exhaustive search over orderings, and the search does not finish past about twenty panels.

Nothing has ever been taught about the second, and it is where most of what goes wrong goes wrong.

The size of the gap

The square twist is small enough to count exactly. Four thousand and ninety-six labellings; two hundred and fifty-six pass every vertex condition; eight have a folded state.

The refusal the letters can see, and the one only a search canSix developable quadrilateral meshes, each asked twice whether its panels can be stacked: once by reading the arcs its letters force, and once by searching every ordering. The letters agree with themselves on all six; the search refuses four of them.the bar is the nodes the ordering search visitedthe letters are consistent on every one of these, so the one-pass test says nothing about any of themmesh 37,4739 panels · 7,473 nodes · no order existsmesh 58,0079 panels · 8,007 nodes · no order existsmesh 89,3469 panels · 9,346 nodes · no order existsmesh 111,0159 panels · 1,015 nodes · an order existsmesh 141449 panels · 144 nodes · an order existsmesh 199,0629 panels · 9,062 nodes · no order existsa red bar is a pattern with no folded state, found only by visiting every ordering it might have had
Fig. 1 The square twist sieved by the two tests that can be applied to a labelling on its own: the letters agreeing with themselves, and the panels admitting an order. The first is cheap and the second is not, and the gap between the two counts is the whole of what a beginner is not told.

So the four teachable conditions remove ninety-four per cent of the labellings and leave two hundred and forty-eight impossible ones standing. A student who has learned everything the subject teaches, applied it perfectly, and drawn a pattern that satisfies all of it has a ninety-seven per cent chance of having drawn something that does not fold.

That is the state of the pedagogy, and it is not the teachers’ fault. There has been nothing to teach.

What the two audiences were told

The gap has a history and it is a history of two audiences that never met.

Froebel’s kindergarten taught folding as geometry to children, and its geometry is about the shapes the folds make rather than about the folds themselves — there is nothing in it about which layer goes where, because the models are small enough that the question never arises.

The mathematical treatment arrived a century later and it is about vertices. Kawasaki’s condition, Maekawa’s count, the big-little-big lemma: all four are statements about a neighbourhood, all four are provable in a page, and all four fit a lecture. The layer question is in the same literature, and it is in it as a hardness result — as a statement about what cannot be done rather than as a technique.

So one audience got a craft with no theory and the other got a theory about points. Neither was given anything about layers, because there was nothing to give that was not either a search or a proof of difficulty.

What folders use instead

Something is plainly being transmitted, because experienced folders very rarely produce impossible patterns, and what they have is not a rule but a set of habits.

Work from the middle outward. Put the largest flap underneath. If it will not go, unfold one step and try the other order. Those are search strategies for the layer ordering, learned by hand, and they are effective enough that a person who has folded for years develops a reliable feel for which arrangements are available.

They are also unwritten, unfalsifiable, and untransferable except by sitting beside somebody. The half no notation records is the same absence seen in the documents: every notation the subject has records the pattern and none of them records the order, so the knowledge has nowhere to live but in hands.

The half no notation writes downFor each pattern this site prints, the size of what a crease pattern records — its vertices, edges and letters — against the number of bits it would take to say which ordering of its panels the folded object is. Every notation the subject has records the first. The interchange format has a field for the second and nothing fills it in.what is recorded, against what is left to the folderThe preliminary base8 panels · 15 bits of orderThe Miura fold24 panels · 79 bits of orderThe square twist9 panels · 18 bits of orderThe hexagon twist13 panels · 33 bits of orderThe Yoshimura pattern65 panels · 302 bits of orderFold and cut — the triangle7 panels · 12 bits of orderThe tapered corrugation28 panels · 98 bits of orderThe waterbomb tessellation52 panels · 226 bits of orderthe pattern, as every format records itthe order of the panels, which none of them does
Fig. 2 For each printed pattern: what a crease pattern records against the bits it would take to say which ordering of its panels the folded object is. The second is most of the object and none of it is written down anywhere.

The first rule that fits on a finger

A chain of panels whose arrows all point the same way is a proof that no ordering exists, and it is the first thing about layers that can be traced by hand on a printed sheet.

The procedure is short enough to state completely. Pick a crease. It joins two panels; the letter and which side is showing tell which of them ends up on top. Draw an arrow from the lower to the higher. Do it for every crease. Now follow the arrows: if they ever return to a panel already visited, the pattern does not fold, and the panels along the way are the reason.

The letters send the panels round in a circleOne arrow per crease, drawn from the panel that must lie below to the panel that must lie above. The direction is decided by the letter and by whether the near panel has been turned over, so the whole picture is read off the crease list without placing a single layer.each arrow points from the lower panel to the higher one9 panels · 12 creases · 12 arcsa loop of 8 panels — no order existsthe arrows are the whole of the test — nothing here asks which panels lie over which
Fig. 3 The square twist with a uniform central ring, with one arrow per crease. Eight of the nine panels are in a circle, and following it takes about ten seconds with a finger.

That is a genuinely teachable rule. It requires no search, no computation and no equipment; it is checkable on a sheet of paper by a person who has been shown it once; and when it fires it hands back a specific chain of panels rather than a verdict.

It is also very far from complete. On the square twist it catches four of the two hundred and forty-eight failures, and on a fold-and-cut pattern it catches none. Consistent is not foldable is the measurement, and it is not flattering.

What it costs to be wrong

The reason this matters more in folding than the same gap would matter elsewhere is that the cost of a wrong pattern is paid in an evening.

A student handed an impossible pattern does not get an error message. They get a sheet that goes together for twenty minutes and then will not close, and the natural conclusion — the one the craft’s whole culture supports — is that they folded it badly. There is no way to distinguish this pattern cannot be folded from it has not been folded well enough without either the search or somebody more experienced.

That asymmetry is why a partial rule earns more here than its coverage suggests. A rule that fires on one failure in sixty is still a rule that occasionally converts an evening of frustration into a ten-second check, and more importantly it establishes that the pattern can be at fault at all, which is the piece of the picture the craft’s framing quietly denies.

The loop is short and the tangle it lies in is half the sheetA tessellation patch with every panel that lies on some loop of the forced order shaded. The cycle a search reports is a dozen panels; the set of panels that could be on one is most of the patch, which is why removing a single crease never repairs it.shaded is every panel that lies on some loop49 panels · 1 tangle · biggest 3535 panels on some loop — 71.4% of the patch52 of 84 arcs run inside it, so one cut removes one of them
Fig. 4 What a failure looks like when it is the pattern’s fault: a tessellation patch with every panel that lies on some circle shaded. Thirty-five of forty-nine, and no amount of care with the paper touches any of it.

One and a half per cent is the wrong denominator

The coverage figure that makes the rule sound negligible is measured over the wrong population, and correcting the population changes the number by two orders of magnitude.

Four of two hundred and forty-eight is the share of the square twist’s admissible labellings that the chain test catches — a uniform count over a set nobody draws from. What a person actually produces is a labelling reached by propagating the conditions and deciding where they stall, and that is a different distribution.

Measured that way, the fire rate is much higher. Of two hundred independently drawn labellings, the test fires on about one in ten for a Miura, one in twenty for a Yoshimura, and seven in eight for a square tessellation patch. On a rhombille patch it fires on all of them.

So the rule catches a tenth of what a person drawing a Miura would draw, and nearly everything a person drawing a tessellation patch would draw. That is a rule worth ten seconds.

And its coverage has a rule of its own

Better still, the test comes with a cheap predictor of when it can fire at all, so a student can be told where to bother.

The test needs a closed chain of panels, and the number of independent chains a pattern has is its count of interior vertices. So: a pattern whose creases radiate without closing — a single vertex, a base, a fold-and-cut outline — has no chain the test can fire on, and walking the arrows there is guaranteed to learn nothing. A pattern with many interior vertices has many chains, and the chance of a drawn labelling closing one rises with the count.

That gives a two-line teaching rule rather than one. Count the interior vertices. If the creases never enclose one, the test cannot help. Otherwise walk the arrows, and the more vertices there are the more likely it is to fire.

The coverage rises exactly where the difficulty does, which is a happier alignment than a partial rule usually gets. The patterns where the test is vacuous are the ones a beginner can fold by trying; the patterns where it fires on seven letterings in eight are the ones nothing else in the subject can say anything about at all.

Why partial rules are worth teaching anyway

A rule that catches one and a half per cent of the failures sounds not worth the breath, and the objection misreads what a partial rule does.

The four vertex conditions are partial too. They catch a great deal, they are necessary and not sufficient, and nobody proposes dropping them on the grounds that a pattern can pass all four and still fail. Their value is that they are cheap and always applicable: they never cost anything, and where they fire the answer is settled.

The chain test has the same shape one level up. Where it fires the answer is settled, and it costs a walk. Where it does not fire, nothing has been learned and nothing has been spent.

No lettering of one vertex puts its panels in a loopEvery mountain-valley labelling of a single interior vertex, at three degrees, counted twice: how many satisfy every condition the subject has, and how many of those force a directed loop among the panels round the point. The second count is zero at every degree.the bar is the letterings that pass every condition at the vertexnone of them forces a loop, because the one lettering that would is the one Maekawa forbidsdegree 48 pass · 0 loop16 letterings · 8 admissible · the alternation fails Maekawa alonedegree 630 pass · 0 loop64 letterings · 30 admissible · the alternation fails Maekawa alonedegree 8112 pass · 0 loop256 letterings · 112 admissible · the alternation fails Maekawa alonechecked at equal sectors and at a skew of 0.18 radians, so the count is not a fact about a symmetry
Fig. 5 Every labelling of a single vertex at three degrees, with how many force a chain: none, at any degree. So the chain test can never fire at a point, which is exactly why it adds something the four conditions do not.

What makes it worth teaching in particular is where it fires. It fires on patterns with closed circuits of creases — twists, tessellations, anything periodic — which are exactly the patterns a beginner meets when they move past single-vertex exercises and try something ambitious. And it fires on the labelling a beginner draws.

The labelling a beginner draws

Taught with a wrong reason collects three cases where the received explanation is wrong and its conclusion right, and two of them are labellings that suggest themselves and do not work: four mountains and four valleys on the preliminary base, which Maekawa forbids; a twist’s central ring reading as one letter, which no folded state admits.

The chain test speaks to the second of those directly. A ring that reads as one letter closes the chain going round the twist, and the eight panels involved are the ring. So the natural mistake now has a demonstration attached rather than an authority: not that lettering does not work, but follow the arrows round and see where they arrive.

Mountain, valley, mountain, valley — and the theorem that refuses itA single interior vertex lettered by the rule every beginner is taught: alternate the letters round the point. The three conditions are evaluated on it. Two of them hold; the counting one does not, and it is the same lettering that would have closed a loop among the panels.the letters alternate strictly round the pointwhich is the only lettering that could send every arc the same way round the panels4 creases · 2 mountain · 2 valleyKawasakiholdsthe sectors alternate to zerobig-little-bigholdsthe smallest sector's two creases differMaekawarefused2 mountains and 2 valleys — the difference is 0, not 2the pattern is drawn without verification, because the point of it is that it does not fold
Fig. 6 The labelling a beginner actually draws, and the theorem that refuses it. Alternating the letters round a vertex is the tidiest thing a hand can do and is exactly what cannot fold, which is why the first rule anyone is taught is a prohibition.

The first case — the preliminary base — the chain test says nothing about, because Maekawa has already refused it and there is no chain to walk. Which is the right division of labour: the counting theorem handles what happens at a point, and the chain handles what happens round a circuit, and a student now has one rule for each.

What makes a rule teachable

Three properties, and the chain has all three where nothing else about layers has any.

It is local in the doing, even though what it establishes is global. Each step looks at one crease and its two panels; no step requires holding the whole pattern in mind. That is what separates a procedure a person can follow from a computation they can only trust.

It is falsifiable by the learner. Having found a circle, a student can go and try to fold the pattern, and it will not fold. A rule whose verdict can be checked against paper in four minutes is a rule that teaches itself.

And it hands back a reason. Not this does not fold but these eight panels each have to be under the next, which is a sentence a person can repeat and can point at on the sheet.

Doing it on a printed sheet

The procedure is worth setting out at the level of a person and a piece of paper, since that is the claim.

Take a printed twist. Number its panels — nine of them on a square twist, and they are the regions the creases divide the paper into. For each crease, decide which of its two panels ends up on top: a valley brings the far panel over the near one, a mountain takes it under, and if the near panel has itself been turned over on the way then the answer swaps. Mark the arrow on the paper.

The turning-over part is the only step that takes thought, and it has a shortcut: the panels two-colour, so colour them alternately as a first pass and the orientation is read off the colour.

Then follow arrows. On a nine-panel twist a circle turns up in a few steps or does not turn up at all.

On a patch of forty-nine panels this is no longer a hand procedure — eighty-four arrows and a walk through them is a computer’s job — but a folder does not need to do it there. What they need there is the knowledge that most letterings of such a patch fail, which is a different kind of thing to teach and equally absent from the literature.

Two things a beginner is told that are worse than nothing

Since this is a rung about teaching, two received pieces of advice deserve naming.

If it will not fold, the folder has made a mistake. This is true most of the time and is unfalsifiable as stated, and it forecloses the possibility that the pattern is at fault. The measurements here say that possibility is large: on the square twist, ninety-seven per cent of the labellings that pass every published condition do not fold.

A crease pattern contains everything a folder needs. It does not. It contains the drawing and the letters, and the layer order is most of the object on anything past a few panels. A folder working from a pattern is reconstructing the missing half by search, and telling them it is all there is telling them their search is retrieval.

Neither piece of advice was invented to mislead; both are compressions of something true about the small models they were formed on. They fail at the size at which people now fold.

What is still not teachable

Everything else, and it is worth being blunt about how much that is.

The two hundred and forty-four labellings of the square twist that the chain test misses fail because of where the paper lands: a panel lying between two panels a crease joins, or two folds in the same place interleaving. Both are statements about overlaps, both require the folded state to have been computed, and neither has a form a person can check by looking at a crease pattern.

The two rules the vertex conditions cannot seeBoth non-crossing conditions on a folded stack, drawn in cross-section. Neither is visible to Kawasaki or Maekawa, because both are statements about which layer lies above which and the vertex conditions look only at angles and letters at a single point.taco-tacoallowedforbiddentwo folds at the same place may nest or stand clearthey may not interleavetaco-tortillaallowedforbiddena flat layer may pass outside a foldit may not pass through onea crease pattern can satisfy every vertex condition and still break one of these
Fig. 7 The three rules an ordering must satisfy, drawn on a folded pattern. The first is a relation between two panels and has a cheap test; the other two are statements about three and four panels, and have none.

Nor is there any prospect of one. A short rule for those would be close to a fast algorithm for a hard problem, and the field’s own habits reflect the absence: the reader is expected to decide far more than the notation ever says, and this is the largest of the things left to them.

What the subject teaches instead of this

It is worth naming what does get taught about layers, because it is not nothing and its shape is revealing.

Diagram sequences teach layer order procedurally: step fourteen tucks this flap behind that one, and the reader who follows the steps ends up with the right pile without ever being told what made it right. That is an extremely effective transmission method — it is how the entire craft moved for two centuries — and it transmits no general knowledge whatever. A folder who has made a hundred models by diagram knows a hundred orderings.

The mathematical literature teaches the conditions an ordering must satisfy: the two non-crossing rules, stated precisely, provable in a paragraph each. Those are genuinely general and they are not a test, because knowing what an ordering must satisfy does not tell anybody whether one exists.

Between the procedure and the conditions there is nothing, and the chain is the first thing to occupy that space: general like a condition, checkable like a procedure.

What a course would look like now

Three things, in order, and the third is new.

The conditions at a vertex. Kawasaki, Maekawa, the big-little-big lemma, developability. Necessary, local, cheap, and taught already wherever the mathematics is taught at all.

That they are not enough. Local is not global needs to be said explicitly rather than left as an advanced topic, because a student who thinks the four conditions decide the question will draw impossible patterns confidently. The square twist’s numbers are the demonstration and they take a minute.

The chain. One arrow per crease, follow them round, and a circle is a proof. It is the only thing in the subject that says no about layers on evidence a person can hold.

How often a redrawn lettering is consistent with itselfIndependent letterings drawn from each pattern, and how many of them the letters do not contradict. A pattern this site prints is nearly always consistent whatever letters it is given; a tessellation patch cut from the same construction almost never is.the bar is the share of draws whose letters agree among themselvesa draw that disagrees is a proof that the pattern has no flat folded state with those lettersthe preliminary base200 of 2008 panels · 8 creases · 0 contradict themselvesthe square twist198 of 2009 panels · 12 creases · 2 contradict themselvesthe Yoshimura190 of 20065 panels · 86 creases · 10 contradict themselvesthe Miura fold181 of 20024 panels · 38 creases · 19 contradict themselvesa square twist patch26 of 20049 panels · 84 creases · 174 contradict themselvesa hexagonal patch2 of 20077 panels · 142 creases · 198 contradict themselvesa rhombille patch0 of 200157 panels · 282 creases · 200 contradict themselvesthe sampler returns solutions rather than a uniform draw over them, so these are shares of what it found
Fig. 8 Two hundred letterings drawn from each of seven patterns, with the share whose letters agree with themselves. Where the bars are short is where the third item earns its place.

Beyond that, honesty. A folder taking on a tessellation is working past what anybody can verify, and the right thing to hand them is not a rule but the fact that there is not one — along with the observation that a sheet of paper settles the question in about four minutes, which nothing else does.

Where this goes next

the bar is the letterings with a folded statethe row is how many times the letter changes going round the central polygonthe ring reads as one letter032 pass every vertex · 28 have no order2 changes round the ring8192 pass every vertex · 184 have no order4 changes round the ring032 pass every vertex · 32 have no ordera twist looks like a twist when the ring reads as one letter, which is why this was never checked Taught with a wrong reason Four mountains and four valleys is what the preliminary base's symmetry suggests and Maekawa forbids it; a twist looks like a twist when its central ring reads as one letter, and no such lettering folds; a tessellation is verified because its unit is, and a forty-nine-panel patch of one had no folded state at all. In each case the conclusion taught is right and the reason offered for it is not, and the site that repeats them is this one. each arrow points from the lower panel to the higher one9 panels · 12 creases · 12 arcsa loop of 8 panels — no order existsthe arrows are the whole of the test — nothing here asks which panels lie over which The ring is the loop The square twist's central polygon is four creases enclosing one panel, and a lettering that gives all four the same letter has no folded state. That was established by enumerating the orderings of nine panels. It can now be read off the crease list in one pass, because the eight panels the letters send round in a circle are exactly the ring — the twist's own defining feature, contradicting itself. the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge The reader decides the junction Five of the eight patterns printed here have places where one crease ends on another — four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle. At each of them a reader has to decide whether two lines meet or pass through one another, and no notation, caption or teaching text in the subject mentions that the decision is being made.

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AssignmentFolded stateKawasaki's theoremLayer orderingMaekawa's theoremNecessary conditionNotationPedagogy