Origami & Folding
what one sheet can do
Flat-folding

The creases that cannot move

One vertex's foldings are always joined up. A pattern's are not, and the number of pieces they fall into is exactly two to the power of the number of creases with an interior vertex at each end — four on a square twist, six on a hexagon twist, none at all on a preliminary base. The creases a local change cannot reach are the creases that never reach the edge of the paper.

Assumes Walking between two foldings and Local is not global.

19 min read 8 figures Local rules, global behaviourThe pattern is the object

The letterings a single vertex folds in are joined up. Change any two creases and any folding becomes any other, on every vertex measured, at every degree the enumeration reaches, on every population tried.

A sheet has more than one vertex, and the obvious guess is that the property survives: repair one vertex, repair the next, and the walk goes through. It does not survive, and it fails immediately — at the smallest pattern with two vertices in it — and the reason is a class of crease that a reader who has thought about the edge of the paper will already suspect.

What a local move cannot reachCrease patterns with more than one vertex, with the letterings their conditions admit sorted into the pieces a local move joins. The count of pieces is exactly two to the power of the number of creases whose two ends are both inside the paper — the creases a folder cannot change without changing two vertices at once.a single vertex is always one piece; a pattern with more is notand the number of pieces is decided by the creases that never reach the edge of the paperThe preliminary base1 vertices inside the paper112 letterings admitted1 piece of 1120 creases buried2^0 = 1The square twist4 vertices inside the paper256 letterings admitted16 pieces of 164 creases buried2^4 = 16The hexagon twist6 vertices inside the paper4096 letterings admitted64 pieces of 646 creases buried2^6 = 64Fold and cut — the triangle1 vertices inside the paper30 letterings admitted1 piece of 300 creases buried2^0 = 1
Fig. 1 Crease patterns with more than one vertex, with the letterings their conditions admit sorted into the pieces a local change joins. A preliminary base is one piece. A square twist is sixteen. The right-hand column is the count that explains both.

What the move is on a pattern

At one vertex the smallest change Maekawa permits is two creases together, and the local version of it is two creases that are neighbours round the vertex.

On a pattern the same move is available and needs saying more carefully: two creases that meet at a vertex, flipped together. Index adjacency in a list of edges is not a move — it joins creases at opposite corners of the sheet, which no pair of fingers can do — so the move is defined by the pattern’s own graph and not by the order the creases were written down in.

That move changes the lettering at one vertex, which is the point. What makes the pattern case different from the vertex case is that a crease has two ends, and flipping it changes the lettering at both of them.

The square twistA square twist: a square with a pleat running out from each of its 4 corners, drawn at 150 mm and carrying 6 mountain and 6 valley creases — 704 mm of folding on a sheet 150 mm across. As the sheet closes the square rotates, which is what gives the family its name. The sector angles are fixed by Kawasaki and the assignment is chosen for having a folded state rather than for reading well — and the unit is verified, while the tessellation it belongs to is not.4 corners, all alikesectors 90°, 90°, 90°, 90°two equal pairs, so no sectoris strictly the smallestthe assignment256 of 4096 fold6 mountain, 6 valleythe ring takes two lettersthe panels can be orderedwhat is checked4 interior verticesand not the tilinga twist of radius 0.17 sheet-widths12 creases, 4.70 sheet-widths of foldingmountainvalleyraw edge
square twist — sheet 150×150 mm — 6 mountain, 6 valley, 704.28 mm of crease
Fig. 2 The pattern the argument is easiest to see on: a square twist, four interior vertices, twelve creases, and a central square whose four sides are the only creases with an interior vertex at each end.

The count, and the rule behind it

A square twist — a polygon that rotates as the sheet closes round it — admits 256 letterings of its twelve creases out of 4,096, one in sixteen. Under the local move those 256 fall into sixteen pieces of sixteen, and every piece is closed: no move takes a lettering in one piece to a lettering in another.

A hexagon twist admits 4,096 of its 262,144 and falls into sixty-four pieces. A preliminary base admits 112 of 256 and is a single piece. A fold-and-cut pattern for a triangle admits 30 of 64 and is a single piece.

Sixteen, sixty-four, one, one. Two to the fourth, two to the sixth, two to the zeroth, two to the zeroth — and the exponent is a count that can be read straight off the pattern.

It is the number of creases whose two ends are both interior vertices. The square twist has four of them: the four sides of its central square. The hexagon twist has six: the six sides of its central hexagon. The preliminary base has none, and neither does the fold-and-cut triangle — every crease of both runs from an interior vertex out to the edge of the sheet.

The last column counts the creases with an interior vertex at each end — the buried ones:

pattern creases admitted pieces buried
preliminary base 8 112 1 0
fold-and-cut, triangle 6 30 1 0
square twist 12 256 16 4
hexagon twist 18 4,096 64 6

The check that makes it a rule rather than a coincidence is stronger than the count: within each piece, the letters on those buried creases are the same in every member, and across the pieces they take every one of their two-to-the-count values exactly once. So a piece is not merely the right size — it is labelled by the buried letters, and the labelling is exact on every pattern the enumeration reaches.

The letterings that fold, and what joins themEvery mountain-and-valley lettering a single vertex folds flat in, joined wherever one small change turns one into another. Whether the picture is one piece or several is the question: a change of any two creases joins all of them, and a change of two neighbouring creases does not.8 letterings fold · 1 piece under any two creasesflip two creases anywhere round the vertex, which is the smallest change Maekawa allowsMMVVVVMMMMVVMVVVMVMVMMMVVMVVVMVMMMVMVVVVMMVVMMMMsectors 43° · 110° · 121° · 57° · 16° · 13°one piece: every folding is reachableevery crease at once: stays inside its own piece
Fig. 3 The contrast, at the object where the answer is different: a single vertex, whose letterings are one piece under the same kind of move. Every crease at a lone vertex runs off to the edge of the paper, so there is nothing buried and the exponent is zero.
The letterings that fold, and what joins themEvery mountain-and-valley lettering a single vertex folds flat in, joined wherever one small change turns one into another. Whether the picture is one piece or several is the question: a change of any two creases joins all of them, and a change of two neighbouring creases does not.8 letterings fold · 2 pieces under two neighbouring creasesflip two creases that are next to one another, which is the change a folder makes by handMMVVVVMMMMVVMVVVMVMVMMMVVMVVVMVMMMVMVVVVMMVVMMMMsectors 43° · 110° · 121° · 57° · 16° · 13°the two pieces are each other turned overevery crease at once: changes piece
Fig. 4 The count, and the rule behind it: the moves available at a degree-six vertex drawn as a graph. Each crease with an interior vertex at both ends is a coordinate no edge of this graph changes.

Why a buried crease cannot move

The mechanism is one sentence and it is worth stating before the consequences.

A move flips two creases that meet at a vertex, and it is allowed because the resulting lettering still satisfies every condition — at that vertex and at every other. A crease with one end on the paper’s edge has only one vertex to satisfy: the boundary end imposes nothing, because no vertex theorem applies there at all. Flipping it disturbs one vertex, and the other crease in the pair is chosen to repair that vertex.

A buried crease has an interior vertex at each end. Flipping it disturbs two vertices at once, and a single move has only one other crease to spend on repairs. Both ends would have to be fixed, the fix at one end is a flip that disturbs a third vertex, and the chain does not close: on every pattern measured, no sequence of moves changes a buried crease’s letter at all.

So the pieces are not a subtlety of the search. They are a conserved quantity — a set of letters that every local operation preserves — and the number of pieces is two to the number of conserved letters.

The chain is worth following once, because the reason it does not close is the reason the whole rung exists. Flip a buried crease and both of its vertices are now wrong. Repair the first by flipping one of its other creases; that crease has a far end, and if the far end is on the paper’s edge the repair is finished there, and if it is another interior vertex the damage has moved rather than gone. On a twist the central polygon’s creases have interior vertices at both ends and so do their neighbours round the polygon, so the damage walks round the ring and arrives back where it started, having changed every letter on the way. That walk is not a move; it is the reversal of a whole cycle, which is a different operation and a much larger one.

What a local move cannot reachCrease patterns with more than one vertex, with the letterings their conditions admit sorted into the pieces a local move joins. The count of pieces is exactly two to the power of the number of creases whose two ends are both inside the paper — the creases a folder cannot change without changing two vertices at once.a single vertex is always one piece; a pattern with more is notand the number of pieces is decided by the creases that never reach the edge of the paperThe preliminary base1 vertices inside the paper112 letterings admitted1 piece of 1120 creases buried2^0 = 1The square twist4 vertices inside the paper256 letterings admitted16 pieces of 164 creases buried2^4 = 16The hexagon twist6 vertices inside the paper4096 letterings admitted64 pieces of 646 creases buried2^6 = 64Fold and cut — the triangle1 vertices inside the paper30 letterings admitted1 piece of 300 creases buried2^0 = 1
Fig. 5 Why a buried crease cannot move, drawn in full: the pieces of the hexagon twist’s folding set. It is the largest pattern whose letterings can be listed at all, and the pieces are exactly the classes the buried creases cut it into.

The one-dimensional case, where the rule is cleanest

A strip has no Maekawa condition — there is no cycle for the folded cross-section to turn through — so its smallest move is a single crease, and the question can be asked without any of the two-crease bookkeeping.

Over sixty strips with creases at random positions, at three to seven creases: the foldings are in one piece exactly when every lettering folds. Not usually, not on average — on all sixty, with no exception. A strip that constrains nothing is trivially connected; a strip that constrains anything comes apart, into two pieces or four or eight.

A strip's foldings come apartStrips with creases at random positions, against how many pieces their flat-foldable letterings fall into when the only move allowed is changing one crease. It is one exactly when every lettering folds and nothing is being constrained; wherever the strip constrains anything the set comes apart, and on some strips every folding is isolated — no single change to any of them stays inside the set at all.345601234creases in the strippieces the foldings fall into5 fold7 fold11 fold26 fold19 of 20 strips constrain somethingand every one of those comes apart1 entirely isolateda strip has no Maekawa condition,so one crease is a move here
Fig. 6 Strips at random crease positions, against how many pieces their foldings fall into under a one-crease change. The rule has no exception in sixty strips: one piece exactly when the strip rules nothing out.

Six of the sixty are worse than split, and the strips that behave are the evenly creased ones — which are the ones every published count is about. Their foldings are entirely isolated — eight letterings fold and not one of them can be changed at a single crease and still fold, so every folding is its own piece. A search on such a strip that begins from a valid lettering and improves it locally is a search that cannot take a step.

A folder’s version of the same fact

The rule can be stated without any of the machinery, and in that form it is something a folder could have noticed.

Take a twist tessellation and try to change it by hand. The pleats can be re-crimped: each of them runs out to the edge of the paper, so pushing one the other way disturbs exactly one vertex and the neighbouring pleat repairs it. The central polygon cannot. Its sides are surrounded on both sides by the pattern, and changing one of them means changing the letterings at two vertices at once — which is not a thing fingers do, because there is no single crimp that performs it.

So the sense in which a twist “has a direction” — the sense in which the central polygon rotates one way rather than the other — is not a property of how the figure was drawn. It is one of the four conserved letters, chosen when the pattern was written down, and no amount of pushing on the pleats will change it. The rotation the family is named for is exactly this quantity.

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 1.000footprint 0.333 · 2.99 layers on average · 9 at the deepest0.333 × 2.99 = 0.995, which is the sheet
Fig. 7 The folded twist, whose central polygon is the region the conserved letters describe. Every pleat around it can be re-crimped by hand; the four creases bounding it cannot, and the folded object shows which way they went.

Which theorem was checked, and how

Nothing above uses a two-dimensional decision procedure, because there is not one. Membership in the set being walked is what the four conditions decide: every vertex passes. That is necessary and not sufficient for the sheet to fold flat, and saying so is the whole of the honesty this rests on — the global question is intractable and the set being walked is the set any local search actually moves through, which is the set the conditions admit.

In one dimension the situation is better and the checks reflect it: the strip solver decides completely, so a strip’s folding set is exactly the foldings.

Three things are asserted rather than assumed.

The move must be the pattern’s own. Two creases are a legal pair when they share a vertex, which is read from the pattern’s edges rather than from the order they were built in.

The piece count must equal two to the buried count on every pattern small enough to enumerate a lettering at a time, and the figure refuses to draw if it does not — so a pattern that broke the rule would stop the page rather than appear on it.

The single-crease move must join nothing at a vertex. It is not a move there, and the machinery must find no edges rather than silently widening the move.

What is proved and what is measured

The rule has two halves and only one of them is an argument, so it is worth separating them before the limits are listed.

The half that is proved is a lower bound. A move flips two creases sharing a vertex, and the paragraph above shows that no move can flip a buried one — so the buried letters are conserved, two letterings that differ in any of them lie in different pieces, and there are therefore at least 2b2^b pieces where bb is the buried count. That much follows from the definition of the move and needs no enumeration at all.

The half that is measured is the equality. That there are no more pieces than that says the non-buried letters are fully navigable: any two letterings agreeing on the buried creases can be walked between. Nothing above proves it, and it is exactly what the enumeration checks, pattern by pattern, on everything small enough to ask.

Keeping the two apart matters because they would fail differently. A pattern with more pieces than 2b2^b would mean a second conserved quantity nobody has identified. A pattern with fewer would mean the argument about buried creases is wrong. The first is the interesting failure and the one to watch for on a larger pattern; the second would be a mistake.

The pieces and their sizes are the same number

There is an arithmetic regularity in the table that the piece count alone does not exhaust, and it is the kind that is either a theorem or a coincidence in two cases.

The square twist has 256 admitted letterings in 16 pieces, so each piece holds 16. The hexagon twist has 4,096 in 64, so each piece holds 64. In both cases the number of pieces and the size of a piece are the same, and both are two to the buried count.

Which means the admitted total is 22b2^{2b} — and on a twist whose central polygon has nn sides, bb is nn and the interior vertices number nn too, so the admitted total is 23n/2n2^{3n}/2^{n}. Two to the crease count, divided by two to the interior-vertex count: each of a twist’s vertices removes exactly half the letterings, which is the same regularity a rectangular map shows, arriving in a family with a completely different shape.

It is not universal, and the same table says so. The preliminary base has one interior vertex of degree eight and admits 112 of 256 — a factor of sixteen sevenths rather than of two, because a degree-eight vertex admits (83)+(85)\binom{8}{3} + \binom{8}{5} of its 282^8 letterings and that is seven sixteenths. The fold-and-cut triangle admits 30 of 64, which is not a power of two either.

So the halving belongs to degree-four vertices and not to vertices in general, and the two twists are the family where every interior vertex is of degree four. That is worth knowing before the rule is carried to a pattern with a mixed degree census, where the factor at each vertex will be that vertex’s own and the product will not be a power of two.

Where the model stops

Eighteen creases is the ceiling. The letterings are enumerated one at a time, so a pattern with more than about eighteen free creases is refused rather than sampled — which leaves the Miura, the Yoshimura, the tapered corrugation and the waterbomb tessellation outside the measurement entirely. The rule holds on every pattern small enough to ask and nothing here says what happens at fifty creases.

A piece is not a folded object. Two letterings in the same piece can still be different folded states, and two in different pieces can produce the same silhouette; the pieces are about what a local change reaches, not about what the paper ends up looking like — which is a distinction the collection has had to make before.

Nothing is claimed about a bigger move. Flipping three creases at once, or flipping every crease of a face, might join the pieces; the measurement is about the move a folder makes and stops there.

Which vertices come apart under a local moveThree populations of vertex, each asked how often the letterings that fold are joined up by a change of any two creases and by a change of two neighbouring creases. The first is always one piece. The second is not, and which population comes apart is the opposite of what a reader would expect.at degree 4, over 16 vertices from each populationthe bar is how many of those vertices stay in one pieceangles at random4 letterings fold at each16 of 1616 of 16multiples of 45°6 letterings fold at each16 of 1616 of 16multiples of 30°4 letterings fold at each16 of 1616 of 16any two creasestwo neighbouring creases
Fig. 8 Where the model stops, counted over the populations: how much of each pattern’s folding set a local move can reach. The buried creases are the part it never reaches, and their share is what decides how many pieces the set has.

What the picture cannot show

The pattern figure is a table, and a table cannot show that the pieces are closed. What is drawn is a count; what is measured is that no move crosses between two of them, which is a statement about every pair of members and therefore about several thousand comparisons that no drawing has room for.

Nor can the drawing show which letterings are in which piece. Sixteen pieces of sixteen is 256 labels, and the useful fact about them — that a piece is named by four letters on four particular creases — is a sentence rather than a picture. The figure that would show it is a picture of a central square with four letters on it, which is a picture of the answer rather than of the evidence.

The idealisation, named

Every crease is a line and every letter is free. On paper a crease that has been folded one way is not free — it remembers, and pushing it the other way costs more than pushing an unfolded one, which is the fourth idealisation and the one that makes the word move a little generous.

The result is unaffected in direction and worth reading with that in mind: paper’s memory adds a cost to every step of the walk, so a walk that is impossible on ideal paper is impossible on real paper too, and one that is possible is merely expensive.

The generalisation

The statement that carries is about inheritance.

A property of every part is not a property of the whole when the parts share their pieces. Each vertex of a twist is connected on its own; the twist is not, and the obstruction is that a crease belongs to two vertices at once. That is exactly the structure a constraint problem has when its variables are shared, and the conserved quantity — the buried letters — is what a solver would call an invariant of the move.

The second half is more useful to a folder. The creases a local change cannot reach are the ones that do not reach the edge of the paper. A pattern whose every crease runs out to the boundary is fully navigable by hand; a pattern with a closed region in the middle of it has a set of decisions that were made when the pattern was drawn and cannot be revisited without redrawing it. A twist tessellation is made of exactly such regions, which is why its pleats are where the interesting arguments happen and its centres are where they do not.

Where the ladder goes next

If a buried crease is what freezes a piece, cutting one should thaw it — and it does, exactly. Cutting any one of the square twist’s four central creases turns two interior vertices into boundary vertices, halves the number of pieces, and doubles the share of letterings the pattern admits.

That is the arithmetic of what a cut buys, and it belongs to a different field’s ladder.

Where this goes next

8 letterings fold · 1 piece under any two creasesflip two creases anywhere round the vertex, which is the smallest change Maekawa allowsMMVVVVMMMMVVMVVVMVMVMMMVVMVVVMVMMMVMVVVVMMVVMMMMsectors 43° · 110° · 121° · 57° · 16° · 13°one piece: every folding is reachableevery crease at once: stays inside its own piece Walking between two foldings The letterings a vertex folds in are always counted and never navigated. Counting says a generic degree-six vertex has eight of them; navigating says that changing any two creases turns any one into any other, and that changing two neighbouring creases does not — and that the vertices which come apart are the ones with no coincidences in them, which is the opposite of what every other measurement here would suggest. MVMV123455 segments, 4 creases12345the stack, solvedassignmentsMVMVvalid stacks1decided byexhaustive searchover the orderingsthe folded positions come from the crease spacing; the assignment only decides which way each turn wraps A strip is decidable Take the same problem down one dimension and it stops being hard. The reason is not that strips are small — it is that overlaps on a line form a chain, and chains cannot contain the cycles that make the two-dimensional question intractable. what the construction produced23 twists, 122 interior verticesturned 24.1° from the tiling's edgespleats 0.068 to 0.068 wide1.58× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge Where two twists share a pleat Every twist tessellation the tradition draws has one size of twist, because every tiling it is drawn on has one kind of vertex. Hand the construction a tiling with two, and the pleat between a large twist and a small one turns out to fix their sizes exactly — three to one, and nothing else folds.

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Boundary vertexCrimpingLayer orderingLocal moveMaekawa's theoremTwist