Origami & Folding
what one sheet can do
Curves and material

One cut removes one arc

A crease pattern whose letters contradict themselves has, in principle, an obvious smallest repair: cut one crease and the statement it was making goes away. Cut every crease of four tessellation patches in turn — four hundred and seventy-four cuts — and sixteen of them leave a sheet whose panels still land anywhere at all. A cut gives the paper a freedom, and a sheet with a freedom in it has no folded state to order.

Assumes A cut is a licence and The loop is not the tangle.

21 min read 8 figures One sheet, no cutsThe pattern is the object

A cut that removes no paper is a cut along an existing crease: the two panels that met along it are still there, still the same size, and the only thing that has changed is that they are no longer joined. In this collection’s terms one edge stops carrying a letter and becomes a raw edge, and everything downstream follows by itself — a vertex at either end now has paper on one side only, so it stops being an interior vertex and every condition being evaluated there stops applying.

A cut is a licence counted what that buys in letterings, and the answer is a great deal. Cutting a crease with an interior vertex at each end releases two vertices from every condition in the subject, and it halves the number of pieces the pattern’s folding set breaks into — so the cut does not merely add letterings, it joins pieces that no local change could get between.

There is a second question about the same operation and the two answers point in opposite directions.

The repair that suggests itself

A crease pattern’s letters can contradict themselves: each crease fixes which of its two panels lies above the other, and a chain of panels each of which must lie below the next is a proof that no order exists. The tangle that chain lies in is most of a tessellation patch, and between forty and sixty-two per cent of the pattern’s arrows run inside it.

A cut removes exactly one of those arrows. It is the smallest change to the layer relation anybody can name — smaller than moving a crease, smaller than changing a letter, since changing a letter reverses an arrow rather than deleting it.

So try it. Every crease, one at a time, on four patches.

A cut does not weaken the contradiction, it dissolves the questionEvery crease of four tessellation patches cut in turn. The bar is how many of those cuts leave a sheet whose panels still place: a cut gives the paper a freedom, so a composition of reflections no longer decides where the far panel goes, and there is no folded state left to ask about.the bar is the cuts after which the panels still place at allnone of them clears the contradiction, and no cut of a buried crease leaves a sheet that placessquare0 of 8484 cut, one at a time · 0 still place · 60 are buried and none of those doeselongated0 of 106106 cut, one at a time · 0 still place · 74 are buried and none of those doeshexagonal14 of 142142 cut, one at a time · 14 still place · 100 are buried and none of those doestriangular2 of 142142 cut, one at a time · 2 still place · 100 are buried and none of those doesa cut along a crease removes no paper — the two panels are still there and are no longer joined
Fig. 1 Four hundred and seventy-four single cuts across four tessellation patches, with how many of them leave a sheet whose panels still land anywhere. Sixteen do. None of the sixteen clears the contradiction.

Sixteen of four hundred and seventy-four, and the four hundred and fifty-eight are not failures to repair. They are sheets that no longer have a folded state at all.

Why the sheet stops placing

The panels of a flat-folded pattern are put where they go by composing reflections. Start anywhere, walk to a neighbouring panel across a crease, and reflect; walk again and reflect again. Where a panel ends up is the product of the reflections along the path, and the whole thing is only consistent because every other path to the same panel gives the same answer.

Cut a crease and that walk loses a step. The two panels it joined are still adjacent — they still share an edge — but crossing that edge is no longer a fold, so nothing about it says where the far panel goes. The paper is free to swing there.

That freedom is exactly what a cut is for: it is why one cut buys a sheet things a fold cannot, and why a slit lets paper reach configurations no crease pattern reaches. It is also why a cut sheet is not a flat folded state in the sense the layer relation is defined on. There is no single arrangement of panels to order; there is a family of them.

A wedge out, and the cone that closesA disc of paper with a 60° wedge marked for removal, and the cone the rest of it closes into when the two cut edges are brought together. Nothing is stretched: the paper that is left is exactly the paper that was there. What changed is how much of it surrounds the centre, and that fixes the cone's half-angle at 56.44° with nothing left to choose.the sheeta wedge of 60° marked for removal60°56.4°what it closes intoa cone of half-angle 56.44°83.3% of the turn is left, and the sine of the half-angle is that same fractionthe circles of latitude are the disc's own, arriving shorter than a flat sheet would needno fold can do this: folding moves paper about and never alters how much of it surrounds a point
Fig. 2 What a cut does to a sheet’s freedom: the angle at a point, which no folding can change and one cut can.

The machinery refuses such a sheet rather than ordering an object that is not determined, and it refuses it by noticing that the panels no longer close: composed one way the far panel lands here, composed another it lands there, and the disagreement is of order one rather than of order a quadrillionth.

It is worth noticing that this is the same refusal the collection has met from the other direction. The placement check exists because a pattern whose panels do not close is not a folded state and ordering one would be ordering something that does not exist. It was written against tessellation patches whose construction was wrong, and it turns out to be the thing that stops a cut being treated as a small edit.

The arcs the letters force, with no circle in themOne 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 arcsno loop — the letters are consistent among themselvesthe arrows are the whole of the test — nothing here asks which panels lie over which
Fig. 3 The arrows a lettering forces, on a pattern whose panels place. Cutting any crease of this removes one arrow and, on most of them, removes the picture.

The creases that cannot be cut are the ones that matter

The sixteen surviving cuts have a property in common, and it is the finding.

Every one of them is a cut at a crease reaching the sheet’s edge. Not one cut of a crease with an interior vertex at each end — a buried crease — leaves a sheet whose panels place. On the square patch, where sixty of the eighty-four creases are buried, that removes sixty of the candidates before anything else is considered.

A wedge out, and the cone that closesA disc of paper with a 60° wedge marked for removal, and the cone the rest of it closes into when the two cut edges are brought together. Nothing is stretched: the paper that is left is exactly the paper that was there. What changed is how much of it surrounds the centre, and that fixes the cone's half-angle at 56.44° with nothing left to choose.the sheeta wedge of 60° marked for removal60°56.4°what it closes intoa cone of half-angle 56.44°83.3% of the turn is left, and the sine of the half-angle is that same fractionthe circles of latitude are the disc's own, arriving shorter than a flat sheet would needno fold can do this: folding moves paper about and never alters how much of it surrounds a point
Fig. 4 Every crease of a small pattern cut in turn, sorted by whether it was buried. The two groups behave differently in every measurement this collection makes of cutting, and this is another.

And the buried creases are where the contradiction is. The letters a local move cannot reach are exactly the buried ones, and the arrows inside the tangle number fifty-two of eighty-four on the square patch against sixty buried creases — so the arrows a repair would have to reach are overwhelmingly on creases a cut cannot touch without destroying the sheet.

That is a considerably sharper statement than the cut does not help. The operation’s reach and the fault’s location are complementary: a cut works at the rim, and the fault is not at the rim.

The sixteen that survive are worth a sentence too, since sixteen is not nothing. They are cuts at creases with an end on the rim — a slit from the edge inward, in effect — and after them the patch still places because the freedom the cut introduces happens to be pinned by the rest of the pattern: the far panel has another route to it that is still all reflections. Whether that happens depends on where in the patch the crease is rather than on anything about the crease itself, which is why it is fourteen creases on one patch and none on another.

A cut does not weaken the contradiction, it dissolves the questionEvery crease of four tessellation patches cut in turn. The bar is how many of those cuts leave a sheet whose panels still place: a cut gives the paper a freedom, so a composition of reflections no longer decides where the far panel goes, and there is no folded state left to ask about.the bar is the cuts after which the panels still place at allnone of them clears the contradiction, and no cut of a buried crease leaves a sheet that placessquare0 of 8484 cut, one at a time · 0 still place · 60 are buried and none of those doeselongated0 of 106106 cut, one at a time · 0 still place · 74 are buried and none of those doeshexagonal14 of 142142 cut, one at a time · 14 still place · 100 are buried and none of those doestriangular2 of 142142 cut, one at a time · 2 still place · 100 are buried and none of those doesa cut along a crease removes no paper — the two panels are still there and are no longer joined
Fig. 5 The same sweep on a different lettering of each patch. The count of surviving cuts moves a little; the count of surviving cuts at buried creases stays at nought.

The two vertices a cut releases

There is one more asymmetry between the two accountings and it explains the rest of them.

Cutting a buried crease releases two interior vertices. Each of them had paper on both sides and four conditions being evaluated there; after the cut each has paper on one side only, and the conditions stop applying. That is the source of everything generous the operation does in the space of letterings: two fewer constrained points means a great many more admissible labellings, and the pieces of the folding set merge because the buried crease is no longer buried.

The layer relation does not care about vertices at all. Its arrows are one per crease and its chains are chains of panels; releasing a vertex changes nothing it reads, except by deleting the one arrow. So the operation’s whole benefit is spent in a currency the contradiction is not denominated in.

How much room a pattern gives its letters to disagreeEvery pattern family here plotted by how many independent closed chains of panels it has against how often an independently drawn lettering agrees with itself. The count is Euler's relation on the panel graph and equals the number of interior vertices; it is read off the drawing before any letter is chosen.more chains is more chances for one of them to closethe printed shelftessellation patchesfold-and-cut outlinessheets folded at random00.2500.5000.7501255075100125independent closed chains of panelsshare of letterings that agree with themselvesa point at nought is nought of the draws taken, which is not a proof that no consistent lettering exists
Fig. 6 The same creases sorted by how much room the pattern leaves its letters to disagree. The ones with no room are the buried ones, which is where a contradiction runs and where a cut is the only thing that reaches.

The generosity is exactly a factor of eight

The two accountings can be priced against each other, and pricing them turns the disagreement from a qualitative one into an arithmetic one.

A pattern with EE creases and VV interior vertices admits about 2E−2V2^{E-2V} letterings: two to the crease count, divided by four at each vertex, since Maekawa halves and the smallest-sector lemma halves again.

Cut a buried crease. The crease count falls by one and the interior vertex count falls by two, so the admissible count becomes

2(E−1)−2(V−2)=2E−2V+32^{(E-1)-2(V-2)} = 2^{E-2V+3}

Exactly eight times as many letterings, from one cut. The lost crease costs a factor of two and the two released vertices buy sixteen.

And it is only available where it destroys the sheet

Now do the same for the cuts that survive.

A crease with one end on the rim releases one interior vertex, so the count goes as 2E−1−2(V−1)=2E−2V+12^{E-1-2(V-1)} = 2^{E-2V+1} — a factor of two. A crease with both ends on the rim releases none, and the count halves.

So the three kinds of cut are worth eight, two and a half, and the sweep says the first kind always destroys the folded state while the other two sometimes do not.

The operation’s entire generosity lives on the creases it cannot be applied to. That is a sharper version of the essay’s conclusion than “the benefit is spent in the wrong currency”: the benefit is a factor of eight, it is exactly the benefit a designer would want, and it is available only at the price of the object.

The sixteen survivors are therefore worth two apiece at best, against sixty candidates worth eight that cannot be taken. Sixteen cuts buying a factor of two each, on patches whose consistent letterings are one draw in eight, is a repair that does not move the number it would need to move.

It also explains the shape of the sweep’s result without needing the sweep. Every crease worth eight is buried by definition — a factor of sixteen requires two interior vertices — and every buried crease takes two panels’ worth of placement with it. The two properties are the same property, so no amount of searching the crease list could have turned up a cut that was both generous and safe.

Two accountings of one operation

Put the two measurements side by side and the disagreement is the point.

In letterings, a cut of a buried crease is generous. It releases two vertices, halves the piece count of the folding set, and joins regions of the space that were unreachable from one another. It is the largest single change to the combinatorics available.

In layers, the same cut is not a change at all. It removes one arrow and takes the folded state with it, so the question the arrow was part of stops being askable.

Neither accounting is wrong; they are about different objects. The first is about the space of labellings a pattern admits, which is combinatorial and knows nothing about where the paper goes. The second is about one labelling’s folded state, which is geometric and is destroyed by the freedom the first is celebrating.

The lesson is the general one about operations that look small: small has to be measured against the thing being preserved, and a cut preserves the crease pattern while destroying the folded state. A cut is not local made the same point about a different consequence.

What a folder would find

The account above is entirely about what a checker can compute, and the thing on the table behaves the same way, which is worth saying because it is not obvious.

Take a patch that has been printed, cut one of its interior creases with a knife, and try to fold it. The pattern is unchanged everywhere else. What has changed is that at one place the paper can be moved without moving anything else — the two panels can slide past one another, or rotate, or lie in either order — and the sheet no longer has a folded form. It has a range of them, and which one a folder ends up with depends on how they held it.

That is not a repair to a fold. It is a different object, and the reason this collection prints no cuts on its ordinary patterns is precisely that the thing a reader is meant to hold together is determined by the crease pattern alone.

What would count as a repair

Three operations are available against a contradiction, and now they can be ranked properly.

Change a letter. This reverses one arrow rather than deleting it, and it keeps the folded state — the panels still place, because a mountain and a valley reflect across the same line. It is the only one of the three that leaves the object intact. Its problem is that a single flip almost always breaks a vertex condition, so the change that is legal is a pair of flips at a shared vertex, and on a tessellation patch there are almost none of those: of two hundred and sixteen candidate pairs on the square patch, none is legal.

Cut a crease. Removes one arrow, and the folded state with it, on every buried crease.

Reletter entirely. Draw a different admissible labelling from scratch. This is the only one that works, and on the square patch about one draw in eight produces a labelling with no contradiction anywhere.

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. 7 The share of independently drawn letterings that agree with themselves. The third operation is a draw from this column, and on the patches it is the only one of the three with a positive success rate.

There is no path of small changes from a bad labelling to a good one. That is not a failure to look hard enough — it is what the freeze on the move graph means, and the cut was the last remaining candidate for a small change.

How many cuts would be enough

The question the sweep does not answer is how many cuts it would take, and there is a reason it cannot be answered the way it is posed.

Removing arrows until a directed graph has no circle in it is a well-defined thing to want and a hard thing to compute — on the square patch it would mean deleting some subset of the fifty-two arrows inside the tangle, and finding the smallest such subset is not something a sweep gets to. But the harder obstacle is upstream: after the first cut there is no folded state, so there is no second graph to remove an arrow from. The sequence stops at one.

That is not a limitation of this collection’s machinery. It is a statement about what a cut sheet is. The layer relation is defined on a determinate folded object; cutting produces an indeterminate one; and the honest thing to report is a refusal rather than a number obtained by pretending the freedom is not there.

There is a version of the question that does have an answer, and it belongs to a different subject. Cutting the sheet into separate pieces and folding each is always possible and always trivial — the limit is a pattern cut along every crease, which is fifty-two loose panels — so the interesting quantity would be the fewest cuts leaving the sheet in one piece and the panels ordered. Nothing here computes it, and stating what it would be a question about is as far as this goes.

The operation that does keep the folded state

For completeness: there is a change to a pattern that removes an arrow and leaves the folded state intact, and it is not a cut.

Reversing a letter keeps everything geometric. A mountain and a valley reflect across the same line, so the panels land in exactly the same places and the placement is untouched; what changes is the direction of that crease’s arrow. So a letter flip is a genuinely minimal edit to the layer relation — it reverses one arrow rather than deleting it, and reversing an arrow can break a chain.

Its problem is elsewhere. A single flip almost always breaks Maekawa at both ends of the crease, so the legal move is a pair of flips at a shared interior vertex, and on a tessellation patch there are none: of two hundred and sixteen candidate pairs on the square patch, not one leaves every vertex condition satisfied.

So the two candidate small changes fail in complementary ways. The letter flip preserves the object and is not available; the cut is available and does not preserve the object.

Where cutting does help

None of this says cutting is useless against layer problems in general. It says it cannot be used as a minimal repair to a pattern one intends to keep as a flat folded object.

Used deliberately it does something the folding cannot. A cut that reaches the edge changes the sheet’s boundary rather than adding one, and a sheet with a slit from the rim has a folded state where the uncut sheet had none — a ring with three creases across it is exactly that case, and one cut from the hole to the rim gives it a folded state with no crease moved and no letter changed.

The difference between that case and this one is which object the cut is asked to preserve. There, the sheet had no folded state and the cut created the freedom that let one exist. Here, the sheet has a determinate placement and the cut destroys 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. 8 Why the ring is the case where cutting helps: every contradiction a lettering can carry runs round an even number of panels, and a ring crossed by an odd number of creases cannot supply one. A single cut from the hole to the rim turns the ring back into a disc and the parity has somewhere to sit again.

So the two answers are consistent after all, and the rule that reconciles them is short. A cut adds freedom. Freedom is what a sheet with no folded state needs and what a sheet with a contradictory one does not — the contradictory sheet has a perfectly good placement and a bad set of letters, and adding freedom to the placement leaves the letters exactly where they were.

Where this goes next

uncut: 4 vertices inside the paper, 6.3% of letterings admittedcut a crease with an interior vertex at each end25.0% admitted2 vertexes released · 4× the share · 4 such creases, all alikecut a crease that already reaches the edge12.5% admitted1 vertex released · 2× the share · 8 such creases, all alikea released vertex is one the four conditions no longer reach, and each is worth a factor of two A cut is a licence What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried. 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 The loop is not the tangle A search that finds a contradiction in a pattern's letters reports the first circle it meets, and on a tessellation patch that is eight to twelve panels of forty-nine. It reads as a local fault. Decompose the same arrows a second way and the set of panels that lie on some circle is thirty-five of forty-nine on the square patch and ninety-nine of a hundred and fifty-seven on the rhombille — which is why the smallest available repair does not reach it, and cannot be tried on most of the creases at all. 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 A cut that removes no paper Cuts in this subject are graded. Take a wedge out and the angle at a point falls by exactly the wedge; take twice as much and it falls twice as far. A hole is not like that. Its effect on what the sheet can do is the same whether it is a tenth of the paper or a ten-thousandth, and it is the same because it is not a quantity at all.

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AssignmentBuried creaseCrease patternDegrees of freedomFolded stateKirigamiLayer orderingRepair