Origami & Folding
what one sheet can do
Flat-folding

A contradiction is even

A crease pattern's letters can demand a circle of panels each of which lies below the next, which is a proof that the sheet has no folded state. Every such circle found here — one thousand one hundred and forty-nine of them, across every family of patterns this collection draws — has an even number of panels in it, and none has four. Both facts are theorems rather than observations, and they come from opposite ends of the subject.

Assumes The sheet has two sides and A proof in one pass.

20 min read 8 figures The pattern is the objectOne sheet, no cuts

Fold a sheet flat and look at it edge on. The paper goes down, turns, comes back, turns again — and at every turn the side facing the reader swaps. That is the whole content of the sheet has two sides: the panels of a flat folded state can be given one of two colours, by which face of the paper points up, and every crease joins panels of different colours.

A graph whose nodes take two colours so that every edge joins different ones is bipartite, and a bipartite graph has no odd closed chain in it. Walk out along any chain of panels and back to where the walk started, and the colour has to return to what it was — which takes an even number of steps.

So every closed chain of panels in a flat-foldable crease pattern is even, and the count is not a fact about any particular pattern.

Two gaps, two theorems

The length spectrum has a second gap and it is worth separating from the first, because they are produced by different results and each is a different check.

No odd lengths, which is the two-colouring: a closed walk that returns to its starting colour has taken an even number of steps. That is a statement about the panel graph and holds whatever the letters are.

No length four, which is Maekawa. A four-panel circle runs round a single interior vertex, and orienting a chain round one point requires the letters to alternate strictly — which gives equal counts of mountain and valley where a flat-foldable vertex needs a difference of two. So the shortest circle any admissible lettering can produce is six, not four, and six is exactly where the distribution starts.

Two absences, two theorems, one from the colouring of the panels and one from the counting of the letters. Neither is a fact about these patterns.

Which makes the spectrum a self-test

That gives the measurement a property it was not built for and which is worth more than the finding: the length distribution checks the instrument that produced it, twice, for free.

Eleven hundred and forty-nine circles is eleven hundred and forty-nine independent chances for an arc walk to mis-step. A walk that dropped a panel, doubled one, or traversed an arc backwards would produce an odd length, and one odd length in the table would be a defect rather than a discovery. None appeared.

By the ordinary rule for a zero count, that bounds the per-circle error rate at under three in eleven hundred and forty-nine — a quarter of one per cent — and at mean length around ten the per-step rate is a tenth of that.

The second gap tests something else entirely. A four-panel circle would mean a lettering that passes the vertex filter and closes a single-vertex chain, which is Maekawa failing to be enforced. Nought fours is therefore a check on the filter rather than on the walk, run over the same eleven hundred and forty-nine cases at no cost.

So the figure that reports the finding is also the figure that certifies the two instruments behind it, and neither certification had to be written. That is an unusually good arrangement, and it is worth naming because the usual complaint about a measurement like this is that nothing independent confirms it.

What that says about a contradiction

A pattern’s letters can contradict themselves. Each crease says which of the two panels it joins lies above the other, and a circle in those statements — a panel below a panel below a panel, closing on itself — is a proof that no order of the layers exists.

That circle is a closed chain of panels. So it inherits the parity for free: every contradiction has an even number of panels in it, and a circle of nine would not be a surprising discovery about folding but a defect in whatever reported it.

Every contradiction has an even number of panels in itThe length of every circle found in the layer relation, over every population of crease patterns here. No odd length occurs, because the panels of a flat-foldable pattern two-colour; and no length of four occurs, because a circle of four goes round one vertex and the counting theorem closes it.the bar is how many circles of that many panels were found726 circles, from 6 panels to 32, over every pattern family measured here4 panels0round one vertex — Maekawa forbids it5 panels0odd — the two-colouring forbids it6 panels21129.1% of the circles measured7 panels0odd — the two-colouring forbids it8 panels21129.1% of the circles measured9 panels0odd — the two-colouring forbids it10 panels8211.3% of the circles measured11 panels0odd — the two-colouring forbids it12 panels9513.1% of the circles measured13 panels0odd — the two-colouring forbids it14 panels304.1% of the circles measured15 panels0odd — the two-colouring forbids it16 panels314.3% of the circles measured17 panels0odd — the two-colouring forbids it18 panels172.3% of the circles measured19 panels0odd — the two-colouring forbids it20 panels141.9% of the circles measured22 panels81.1% of the circles measured24 panels152.1% of the circles measured26 panels71.0% of the circles measured28 panels20.3% of the circles measured30 panels20.3% of the circles measured32 panels10.1% of the circles measuredthe empty rows are not rare cases — they are lengths that cannot occur, and each has its own reason
Fig. 1 The length of every circle found here, over every family of patterns the collection draws: the printed shelf, five tessellation patches, and nine sheets creased by folding them at random. Eleven hundred and forty-nine circles, from six panels to fifty. No odd length occurs, and neither does four.

Eleven hundred and forty-nine circles and not one odd. The rows for five, seven, nine and eleven are not rare cases that a larger sample would populate. They are lengths the paper cannot produce.

The other empty row

The row for four is empty too, and its reason is a different theorem entirely.

A circle of four panels would be four panels in a ring, each joined to the next by a crease. The shortest ring of panels a crease pattern has is the one round a single interior vertex — a degree-four vertex has exactly four panels between its four creases — and no admissible labelling of a single vertex can send its arrows all the way round. Sending them round requires the letters to alternate strictly, an alternation has equal counts, and Maekawa’s theorem says the counts differ by two.

So the shortest circle available is six, and six is the commonest length observed: three hundred and nineteen of the eleven hundred and forty-nine, with eight not far behind at three hundred and twenty-nine. Together those two account for more than half.

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. 2 A circle of eight, on the square twist: the ring of panels round the central polygon, which is the shortest circuit that pattern has once its vertices are excluded.

The two closed rows are worth holding side by side because they are so unlike each other. Odd lengths are closed by a colouring — a property of the panel graph that follows from paper having two sides, and that would be true of a pattern with no letters on it at all. Four is closed by a count — a property of the letters at a point, established in the eighteenth century by an argument about a cross-section. Neither knows about the other, and the layer relation is where they meet.

Six, and where six comes from

The peak at six is worth pursuing, because a length that occurs three hundred and nineteen times is describing a structure rather than an accident.

A circle of six panels is a closed chain crossing six creases and enclosing more than one vertex — a circuit going round something, with two or three interior vertices inside it. The twist tessellations have exactly that shape built in at every edge of the tiling: two twist polygons joined by a pleat, and the chain of panels running round the pleat from one polygon’s ring to the other’s and back. A pleat is two creases; each polygon contributes two more panels; six.

That is also why the count of six-circles rises so much faster than the patch does. A patch with twice the polygons has roughly twice the pleats, so twice the six-circuits, and each of them is an independent chance for the letters to close one. The bigger the patch, the rarer a lettering that agrees with itself is that observation with a measurement under it.

The bigger the patch, the rarer a lettering that agrees with itselfThe same twist construction over five tilings, ordered by how many panels the folded patch has, against the share of independently drawn letterings whose letters do not contradict themselves. The share falls to nothing well before the patch is large enough to be interesting.the bar is the share of draws that agree with themselvesthe rows are ordered by panel count, which is the only thing changing along them49 panels26 of 200square · 84 creases · 26 of 20062 panels5 of 200elongated · 106 creases · 5 of 20077 panels2 of 200hexagonal · 142 creases · 2 of 20083 panels0 of 200triangular · 142 creases · 0 of 200157 panels0 of 200rhombille · 282 creases · 0 of 200a zero is a zero of the draws taken and not a proof that no consistent lettering exists
Fig. 3 The five patches by panel count against the share of letterings that agree with themselves. The mechanism is in the previous figure: more polygons is more pleats, and more pleats is more short circuits to close.

The printed corrugations are the contrast. A Miura fold’s panels are a grid and its shortest circuits go round its degree-four vertices, which Maekawa has closed; the next shortest go round a whole cell of the grid, and there the arrows have four creases’ worth of freedom to disagree in. It produces circles of eight to fourteen, and it produces them in nineteen draws out of two hundred.

Why this is a check and not a finding

The temptation with a result like this is to present it as a discovery about crease patterns. It is better used as an instrument on the instrument.

Parity is exactly the sort of thing an implementation gets wrong. The direction of an arrow depends on the letter multiplied by whether the near panel has been turned over, and a sign error in that product would produce arrows that are individually plausible and collectively wrong. A wrong sign on one crease of a chain flips the chain’s parity — so a bug of that shape shows up as an odd circle, immediately, on the first pattern that has one.

That is why the parity is checked rather than merely observed. Every circle the machinery returns is tested for evenness where it is produced, and an odd one raises an error naming the two-colouring rather than being reported as a circle of nine panels. In eleven hundred and forty-nine circles it has never fired, which is the correct outcome for a check of this kind: it costs nothing, it would have caught the error it was written for, and it says so every time it does not.

Two colours, and no choice about themThe panels of a flat-foldable pattern, painted in the two colours the creases force on them. The colouring is computed by crossing creases and counting; the parity of the crease counts at the vertices is computed separately from the edges. They are the same fact twice.12 panels, two coloursno crease has the same colour on both sidesall 6 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge
Fig. 4 The two-colouring on a Miura fold’s panels: neighbouring panels never share a colour, because the crease between them is where the paper turns over. Every closed chain in this graph is even, and the layer relation’s circles are drawn from the same chains.

The same reasoning applies one level up. Even is not enough established the other half of the two-colouring’s story — that a pattern’s panels can two-colour while the pattern has no folded state at all, so the colouring is necessary and nowhere near sufficient. Parity here is the same kind of statement: it constrains what a contradiction can look like and says nothing whatever about whether there is one.

A parity that is not this one

There is a second parity argument in this subject that lives one level down and is easy to confuse with this one, so it is worth separating them explicitly.

Maekawa’s count is about the letters at a vertex and its parity statement is that the degree is even: an odd number of creases cannot meet at an interior vertex of a flat-foldable pattern, because the mountains and valleys have to differ by two and their sum has to be the degree. Nothing meets at three is that argument.

The two-colouring is about the panels of the whole pattern and its parity statement is that every closed chain of panels is even. The first is local and about creases; the second is global and about panels; and neither implies the other. A pattern can have every vertex of even degree and no two-colouring at all, which is exactly what a ring with three creases across it is: three is odd, but the vertices are on the boundary and the count never applies to them.

The distribution, and what it is not

Beyond the two closed rows the lengths fall off steadily: a hundred and twenty-seven circles of ten, a hundred and forty-one of twelve, fifty-nine of fourteen, and a long thin tail out to a single circle of fifty on the rhombille patch.

It is tempting to read that shape as telling something about how contradictions are distributed. It is not, and the reason is the same one that makes the length of a reported circle a poor measure of the fault: what is being counted is the first circle a depth-first walk happens to meet, which is biased hard toward short ones. A tangle of thirty-five panels usually has a six-panel circle somewhere in it, and that is what gets reported. The tail is not the shape of the contradictions; it is the shape of what the shortest circle in each of them was.

Every contradiction has an even number of panels in itThe length of every circle found in the layer relation, over every population of crease patterns here. No odd length occurs, because the panels of a flat-foldable pattern two-colour; and no length of four occurs, because a circle of four goes round one vertex and the counting theorem closes it.the bar is how many circles of that many panels were found412 circles, from 6 panels to 30, over every pattern family measured here4 panels0round one vertex — Maekawa forbids it5 panels0odd — the two-colouring forbids it6 panels12730.8% of the circles measured7 panels0odd — the two-colouring forbids it8 panels12630.6% of the circles measured9 panels0odd — the two-colouring forbids it10 panels4711.4% of the circles measured11 panels0odd — the two-colouring forbids it12 panels5012.1% of the circles measured13 panels0odd — the two-colouring forbids it14 panels143.4% of the circles measured15 panels0odd — the two-colouring forbids it16 panels163.9% of the circles measured17 panels0odd — the two-colouring forbids it18 panels102.4% of the circles measured19 panels0odd — the two-colouring forbids it20 panels51.2% of the circles measured22 panels41.0% of the circles measured24 panels61.5% of the circles measured26 panels41.0% of the circles measured28 panels10.2% of the circles measured30 panels20.5% of the circles measuredthe empty rows are not rare cases — they are lengths that cannot occur, and each has its own reason
Fig. 5 The same census over fewer draws. The two closed rows are still empty, which they must be; the tail is shorter, which is what a smaller sample of a quantity with a long tail does.

What the distribution does say, and says usefully, is that short circles are available almost everywhere. If contradictions typically closed at twenty panels, the six-and-eight peak would be absent — a walk cannot report a circle the graph does not have. Their dominance means the circuits carrying the failure are small ones: two twist polygons sharing a pleat, or one twist polygon’s own ring.

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. 6 The seven patterns whose letterings the lengths are drawn from, with the share of draws in which the letters agree. The three patches at the bottom supply most of the circles above them.

What a folder can do with it

A parity is a poor tool for building anything and a good one for ruling things out quickly, and there is one place here where it does real work by hand.

A reader looking at a printed patch and wondering whether a particular lettering will fold has, in principle, to check every closed chain in it. Parity halves that: only the even chains can carry a contradiction, and in a twist tessellation the even chains are the ones going round a pleat or round a polygon, while the odd ones — a chain through a polygon’s ring and out across a single crease and back — are not chains at all in the sense that matters, because they cross a crease an odd number of times and therefore do not close.

That is not a large saving on a computer and it is a considerable one on paper. It is also the answer to a question a folder asks constantly and this collection has not addressed directly: which parts of a pattern have to agree with each other? The chains, and only the even ones.

Where the parity comes from, said carefully

One step of the argument deserves more than the sentence it usually gets, because it is where a reader could reasonably object.

The two-colouring is a statement about the crease pattern, before anything is folded: colour a panel by whether an even or an odd number of creases separates it from a chosen starting panel. That is well defined exactly when every closed chain of panels crosses an even number of creases — which is the same statement as the graph being bipartite, so it looks circular.

It is not circular, because there is an independent reason. A flat folded state assigns each panel a rigid motion of the plane, built by composing one reflection per crease along a path from the starting panel. A reflection reverses orientation. So a panel reached by an odd number of creases is face down and one reached by an even number is face up, and the answer cannot depend on the path, because the panel is somewhere definite in the folded object and is either face up or face down. Any closed chain therefore has an even number of creases in it.

The independence is real: the argument runs through the geometry of the folded state rather than through the combinatorics of the graph. A pattern whose panels do not close — where two paths to the same panel disagree about where it goes — is precisely a pattern for which that argument fails, and such a pattern is refused before any of this is asked of it.

The same parity, with nowhere to put itThe same creases on a square of paper and on a loop of paper. On the left they meet at one interior vertex, which carries the parity and which every theorem in the subject inspects. On the right the middle has been removed, that vertex is gone, and the parity is still there — in the panels, where nothing local can see it.a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart
Fig. 7 The exception that proves the mechanism: a sheet with a hole in it has closed chains of panels that go round the hole rather than round a vertex, and on an odd number of creases the colouring has nowhere to sit.

And there is one shape where it genuinely fails. A ring and a line is a sheet with a hole and an odd number of creases running from the hole to the rim: the chain of panels going round the hole crosses an odd number of creases, so no two-colouring exists, and the parity argument here has no purchase. Such a sheet has no flat folded state either, which is the same fact arriving first.

The one place an odd circle would be legitimate

There is a construction here whose panels do not two-colour, and it is worth asking what the layer relation does on it, since the parity argument has no grip there.

A sheet with a cut in it is the case. A cut is not local established that separating the paper along an edge with paper on both sides is a sixth kind of assignment — not a mountain, not a valley, not a raw edge — and that it removes an adjacency from the walk over panels rather than adding one. Removing an adjacency cannot create an odd chain: it deletes chains, it does not lengthen them. So a cut sheet’s circles are still even, and for a reason that is one step removed from the colouring.

A sheet with a hole is different, and it is the genuine exception. Going round the hole is a closed chain that no vertex accounts for, and on an odd number of creases it is odd. Such a sheet has no flat folded state, so it never reaches the layer relation at all — the placement fails first, and the failure is reported as panels that do not close rather than as a circle. The parity check would fire on it if it ever got that far, and the fact that it never does is the machinery refusing in the right order rather than the parity being vacuous.

What it buys

Three things, in ascending order of usefulness.

It is a sanity check on every circle reported, at no cost, testing exactly the arithmetic most likely to be wrong.

It is a shape constraint on what a contradiction can be, which narrows the search when one is being looked for by hand. A reader tracing arrows on a printed patch knows that a promising-looking chain of seven panels is not a circle and has miscounted somewhere.

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. 8 The square patch with every panel on some circle shaded. Whatever the shape of that region, every circle inside it is even and none of them is four — which constrains the tangle without saying anything about how large it is.

And it puts the two closed rows of that first figure side by side, which is the part worth carrying. The lengths a contradiction cannot have are settled by a colouring that knows nothing about letters and a count that knows nothing about geometry. Everything between six and fifty is the pattern’s own business; the two ends are the subject’s.

Where this goes next

20 panels, two coloursno crease has the same colour on both sidesall 12 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge The sheet has two sides Read a crease pattern as a set of panels rather than a set of lines and a condition appears that no vertex theorem states: the panels take two colours, no crease has the same colour on both sides, and the colour is which face of the paper each panel ends up showing. 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 The loop a vertex cannot close A crease pattern's letters can contradict themselves, and the contradiction is never local. Enumerate every mountain-valley labelling of a single interior vertex at degree four, six and eight — a hundred and fifty pass every condition the subject has — and not one of them sends its panels round in a circle. The one labelling that would is refused by Maekawa, alone: Kawasaki holds on it and so does the big-little-big lemma. a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart Even is not enough Every vertex theorem in the subject is a statement about one point, and the two-colouring of the panels looks like the exception. It is not — on a square of paper it is a parity at each vertex and nothing more. Cut a hole and the two come apart: a loop of paper with three creases has no interior vertices at all, satisfies every theorem there is, and cannot be folded flat.

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AssignmentFace graphFlat-foldabilityFolded stateLayer orderingMaekawa's theoremNecessary conditionTwo-colourability