The Miura folds two ways

One vertex, repeated is what a Miura is, and the repetition is the whole of why it gets built. Identical panels, identical creases, one crease family running straight through every vertex, and — as a sheet with one freedom established — a single degree of freedom, so one input drives the whole sheet.

That last fact is usually stated as though it settled the shape. One freedom, one number, one configuration. It does not.

What a degree of freedom does and does not say #

A mechanism with one degree of freedom has a one-dimensional set of configurations. That is a statement about the dimension of the set and not about its shape, and a one-dimensional set can have several branches — several curves through configuration space, each of them a valid motion, all of them passing through the same values of any single coordinate.

A degree-four vertex is the smallest example. Fix one of its fold angles and there are two configurations, not one: the spherical four-bar it amounts to has two assemblies wherever it has any. That is standard and this site has drawn it since the rigid ladder began.

What has not been asked is what happens when a sheet is made of many such vertices. Each of them offers two, and the choices have to agree on every shared crease, so the count for the sheet is not two to the power of the vertex count — most combinations are killed immediately. The question is how many survive.

Enumerating them #

The walk is exhaustive over branch choices rather than a search that stops at the first answer. Vertices are visited in order; each is solved from a crease it already knows; both of its configurations are tried; and a configuration is kept only if it agrees to a part in ten million with every crease the vertex shares with one already solved.

Two survive on a four-by-four Miura at 0.9 radians. Their measurements:

Half the width. The same height, because the difference is entirely in how the columns stack rather than in how the rows do. And a different letter on three of the twenty-four creases, which is what makes the second state a different folding rather than a different position of the same one.

The second state is not an artefact of the enumeration. It is placed by the same walk, its panels are the same rigid panels, its vertices close to the same precision, and it is a shape a sheet of card could be assembled in.

The mesh that does not do this #

A general quadrilateral mesh — the kind the joint solve produces, with no crease family running straight and no two vertices alike — has one folded state at a given fold angle. Its vertices are all different, so the two configurations at each of them are different pairs, and an accidental agreement between neighbours is exactly what is not available.

That is the comparison the whole rung turns on, and it inverts the usual reading of the Miura. Repetition is normally listed among its virtues: fewer distinct parts, one tool, one inspection. Here the same repetition is what leaves the sheet with more than one answer to a question that ought to have one.

The property that makes a Miura manufacturable is the property that makes it ambiguous.

That sentence has a companion in the tolerance ladder and the two together are uncomfortable. A Miura’s reason for folding survives being cut five per cent wrong and a solved general mesh’s does not survive a third of a millimetre; a general mesh’s shape is determined by one fold angle and a Miura’s is not. The two properties an engineer wants sit on opposite sides of the same structural fact, and no amount of care with either pattern moves them.

What the second state actually is #

It is worth describing the other folding rather than leaving it as a row in a table, because “half the width” is a bigger difference than the phrase suggests and it has a mechanism behind it.

A Miura’s columns run straight through every vertex. In the state everybody draws, the sheet’s zigzag rows nest so that each column of panels leans the same way, and the sheet contracts in both directions together — which is the negative Poisson’s ratio the pattern is famous for. In the second state, one family of columns leans the other way, so alternate columns fold back on themselves rather than nesting, and the sheet closes in that direction twice as far.

The three creases whose letters differ are what implements the change. They are not scattered: they lie along the boundary between the region that has been inverted and the region that has not, which is exactly what a pop-through defect looks like in a sheet somebody has handled roughly.

That is the honest description and it also names the reason the second state has been easy to overlook. It looks like damage. In the geometry it is not damage — it is the other solution of the same equations, and the equations do not distinguish.

How the ambiguity is distributed #

The count depends on which crease is driven. Driving each of a Miura’s twenty-four creases in turn and enumerating leaves one, two, four or eight consistent foldings, depending on the crease.

So the ambiguity is a property of the pair (sheet, driven crease) rather than of the sheet alone. There are creases on a Miura that determine it completely, and creases that leave it with eight shapes to choose between, and nothing about the pattern makes it obvious which is which.

For a sheet that is folded by hand this is invisible: a folder’s hands are on the whole sheet and the shape is decided by which way the paper is pushed. For a sheet that folds itself, it is the whole problem — an actuator on the wrong crease is a sheet that has been told its fold angle and not its shape.

One, two, four, eight #

The counts the survey reports are all powers of two, and that is the whole structure of the ambiguity rather than a coincidence of four numbers.

The second state differs from the first by one family of columns leaning the other way — a band of the sheet inverted relative to its neighbours. A four-by-four Miura has nine interior vertices in a three-by-three arrangement, so it has three such bands, and each can be inverted or not independently of the others.

Three binary choices is $2^{3} = 8$, which is exactly the largest count the survey finds.

And the smaller counts follow. A driven crease lies inside some of the bands and pins them: pin one and four combinations remain, pin two and two remain, pin all three and one remains. The four values 1, 2, 4 and 8 are the four ways a crease can meet three bands, and nothing else is possible.

Which makes it obvious after all #

That answers the essay’s own complaint that “nothing about the pattern makes it obvious which is which”. It is obvious, once the bands are the unit: count how many bands the driven crease belongs to, and the state count is two to the power of the rest.

A crease running along a band’s interior pins one. A crease crossing the sheet pins all of them and leaves the shape determined. So the creases that determine a Miura completely are the ones that run across its columns, and the ones that leave eight shapes are the ones buried inside a single band.

For a self-folding sheet that is an actionable rule rather than a hazard: drive a crease that crosses every band, and the ambiguity disappears without any change to the pattern.

And it says what happens to a larger sheet #

The essay leaves open how the count behaves as the Miura grows, and the band reading answers it in the least comfortable direction.

An $n$-column Miura has $n-1$ interior bands, so a crease pinning none of them leaves

$$2^{\,n-1}$$

states. Four columns give eight; six give thirty-two; ten give five hundred and twelve.

The ambiguity doubles with every column added. It is not a small-sheet curiosity that a larger pattern would wash out; it is exponential in the sheet’s width, and a metre-wide array driven from a badly chosen crease has more consistent shapes than anything could enumerate.

That is a prediction rather than a measurement — it is read off four counts and a mechanism — and it is cheap to test, since the enumeration already exists and a six-column Miura is one run.

The same count on a smaller object #

The result is easier to believe with the one-vertex case beside it, and the one-vertex case is where the two states come from.

A single degree-four vertex with a fixed fold angle on one crease has two configurations. The site has drawn them since the vertex kinematics rung, and the reason they exist is a closure that is quadratic rather than linear: a spherical four-bar assembles two ways or not at all. So ambiguity at a vertex is not a subtlety; it is the ordinary situation, and the surprising thing is that a whole sheet of them usually has one answer.

That is the right way round to hold the result. Ambiguity is the default at a vertex and gets destroyed by the agreements a sheet demands, and a Miura is the case where the agreements happen to be satisfiable both ways.

Which theorem was checked, and how #

Each state is verified three ways, and the third is the one that would catch an enumeration counting the same folding twice.

The angle closure at every vertex is exact: the product of rotations about the four crease directions is the identity to 3 × 10⁻¹⁵.

The placement is an isometry and closes. Every edge and both diagonals of every panel keep the lengths the flat pattern gives them, and the four copies of every vertex — placed by a walk that arrives from different directions and does nothing to make them agree — land on the same point to 2 × 10⁻¹⁵.

And the two states have different letters, not merely different numbers. Three creases that are mountains in one are valleys in the other, so no rigid motion or relabelling takes one to the other, and the difference is not a matter of viewpoint.

The enumeration also refuses one thing outright. A configuration in which some crease is not folded at all is a folded state of a different pattern — the one with that crease rubbed out — and it satisfies every closure exactly. It is excluded, because otherwise the count for every sheet is at least one and the extra is the unfolded sheet.

Why the count is not two to the twenty-fourth #

The enumeration’s shape is worth a paragraph because the answer is so much smaller than the arithmetic suggests.

Nine interior vertices, two configurations each, would be 512 combinations. Almost none survive: a configuration chosen at one vertex fixes all four of its fold angles, and each of those is shared with a neighbour, so the neighbour’s choice is tested immediately against something already decided. The contradictions arrive at the second vertex rather than at the last.

What that means is that the two survivors are not two out of many near-misses. They are two out of a search that is almost entirely pruned, and the pruning is what makes the result a count rather than an estimate. It is also why the general mesh’s answer of one is not surprising once the mechanism is seen: with all vertices different, the agreement that keeps a second branch alive across the sheet has to happen nine times by accident.

Where the model stops #

The count is at one fold angle, on one size of sheet, in one pattern. Nothing here says how the number of states behaves as a Miura is made larger — whether it stays at two, grows, or collapses — and the enumeration’s cost rises quickly enough that the question needs a better method rather than a bigger run.

It is also a count of rigid foldings, which is a stronger requirement than a folded state. The Miura has more flat folded states than this counts, in the sense an earlier rung measured: a marked crease pattern admits several stackings, and stackings are not what is being counted here. What is counted is complete assignments of fold angle to crease consistent with the closure at every vertex, which is a different and smaller collection.

Nothing here says whether a real sheet prefers one state. That is an energy question, it needs a material, and a folded sheet of anything real has a stiffness that would make one of the two easier to reach — which is exactly the mechanism a self-folding sheet uses and is not geometry.

What the picture cannot show #

A picture of two folded states is a picture of two objects, and the thing being claimed is that a single sheet at a single input is both of them until something decides. There is no way to draw an undecided mechanism; the alternatives have to be drawn side by side, and drawing them side by side makes them look like two sheets.

Nor can any of these figures show the flat pattern of the second state, because there is not one: both states have the same crease pattern, with the same creases in the same places. What differs is the letters on three of them, and a letter is a property of the folding rather than of the sheet. Nothing shows the choosing. The transition from one state to the other is not a motion — the sheet cannot travel from one to the other without unfolding, because they differ in letters — so the picture a reader wants, of a sheet moving between them, would be a picture of something that does not happen.

The generalisation #

The right way to state this is about symmetry rather than about origami. A mechanism assembled from identical components has a symmetry group acting on its configuration space, and a symmetry group acting on a configuration space produces multiple configurations at the same input — the orbit of one solution under the symmetry is a set of solutions.

A mechanism assembled from components that all differ has no such group and no such orbit, and its configurations are isolated.

So the trade is completely general and it is not specific to folding: interchangeability of parts buys manufacture and costs determinacy. What origami adds is a case where the trade is extreme — the Miura is as repetitive as a mechanism can be, and it is the pattern the field has settled on for exactly that reason.

There is a second reading worth having. The Miura’s second state is not an obscure configuration reached by a strange path; it is half as wide, which is a difference anybody would notice. A structure that deploys into the wrong one of these has not deployed slightly wrong. It has deployed to half its width.

Who found it, and when #

Multistability in rigid origami is a known subject and the Miura’s alternative folded states are documented in the deployable-structures literature, usually in the context of pop-through defects: a Miura that has been handled roughly acquires patches folded the other way, and the patches are stable. The engineering interest is in preventing them.

What is measured here is the count, from the closure conditions alone, with no material and no energy in it. The two states are not defects in the geometric account — they are the two things the equations allow, and calling one of them a defect is a statement about which one was wanted.

Where the ladder goes next #

The immediate question is which creases determine the sheet. The survey already records the number of foldings each driven crease leaves, and reading the pattern of it — whether the determining creases are a row, a boundary, a diagonal — would turn this rung into an instruction: put the actuator here.

There is also a flat-folding version of the same question waiting. A Miura’s flat folded state is unique up to the stacking, and the ambiguity measured here is about the path rather than the destination — two rigid foldings that reach different shapes at 0.9 radians could still reach the same flat stack at π. Whether they do is a short computation and it would say whether the second state is a different journey or a different arrival.

The other direction is size. Two states on a four-by-four sheet is a small number and a large sheet is what gets deployed, so how the count grows with the number of vertices is the question that decides whether this is a curiosity or a constraint. The enumeration’s cost says the answer needs a different method, which is the usual signal that a rung is worth building machinery for.