A material made of creases

A material has properties. Steel has a stiffness, rubber has a Poisson’s ratio, paper has both and a grain besides. Those numbers are facts about what the stuff is made of, and changing them means changing the stuff.

Fold a sheet and it acquires a new set of them, and the new set has nothing to do with what the sheet is made of.

What Poisson’s ratio measures #

The number is worth restating precisely, because the folded case makes the usual intuition unreliable.

Stretch something in one direction and watch what happens in a perpendicular direction. Poisson’s ratio is minus the transverse strain divided by the applied one. Rubber gives about 0.5, steel about 0.3, cork close to zero — pull cork and it barely narrows, which is why a cork goes into a bottle.

A negative value means the material gets wider when stretched. Very few solids do it. The ones that do are called auxetic and they are mostly foams with re-entrant cell structures, and their negative ratios are a consequence of geometry at a small scale rather than of chemistry.

A folded sheet is the same phenomenon at a scale where the geometry can be drawn — and unlike a foam, its unit cell is a mechanism whose motion can be solved exactly.

The Miura’s ratio is negative and is not a constant #

Take a Miura fold and open it. Both in-plane dimensions grow together; close it and both shrink together. Since the two move the same way, the ratio is negative.

That much is well known. What is less often said is that the value depends on how far the sheet is folded, and depends on it strongly.

At the flat end the ratio is a small negative number set by the panel’s slant. Fold it halfway and the magnitude has roughly doubled. Fold it nearly closed and it runs away — past minus five, past minus ten, and without bound as the sheet approaches its packed state.

So there is no such thing as “the Poisson’s ratio of a Miura fold”. There is a function, and quoting one number from it is quoting a state as well as a pattern.

That is not something a material can do. Steel’s ratio is 0.3 whatever is being done to it, because it is a property of the bonds. A folded sheet’s is a property of the configuration, and the configuration is a variable.

The accordion is the control #

The comparison that makes the point is the boring pattern rather than the interesting one.

Fold the same paper into a plain accordion — parallel creases, alternating — and close it. One dimension shrinks; the other does not move at all, because nothing in the pattern couples them. The ratio is exactly zero, for the whole travel, for every panel width, for every material.

Which settles a question that would otherwise hang over the whole subject: whether the negative ratio is something folding does. It is not. Folding an accordion gives zero, folding a Miura gives a negative number that varies, and the two sheets can be cut from the same page.

So the property belongs to the crease pattern in the strict sense. Same material, same thickness, same grain; different pattern, different mechanics.

The two ratios are reciprocals, and that settles the word #

The runaway at the closed end looks alarming and there is an identity underneath it that makes it ordinary — and that proves the essay’s point more sharply than the variation does.

A sheet with one degree of freedom has both its dimensions controlled by a single parameter, so $dx$ and $dy$ are both proportional to that parameter’s change. Then

$$\nu_{xy} \cdot \nu_{yx} = \left(-\frac{dy/y}{dx/x}\right)\left(-\frac{dx/x}{dy/y}\right) = 1$$

identically, at every configuration, for every one-freedom mechanism. A Miura reading $-10$ measured one way reads $-0.1$ measured the other, and the divergence is a statement about which dimension was called the input rather than about anything the sheet does.

Which no material can satisfy #

That identity is the argument. An isotropic material has the same ratio in both directions, so its product is $\nu^2$ — and $\nu^2 = 1$ only at $\nu = \pm 1$, which steel at 0.3 and rubber at 0.5 are nowhere near.

So a folded sheet does not merely have a ratio that varies where a material’s is fixed. It satisfies a relation no material satisfies at all, and the relation is a direct consequence of having one freedom rather than three dimensions of elastic response.

The accordion is the clean limit of it. Its ratio is exactly zero one way, so the reciprocal is unbounded the other — which is not a defect in the control but the same identity at its endpoint. What makes the accordion a good control is a different property: its zero is constant along the whole travel, and the Miura’s number is not constant along anything.

Which theorem was checked, and how #

Every number in these figures comes from a folding that has been solved rather than posed, and it is worth saying what that means because the previous version of this site’s Miura figures got it wrong.

The folded Miura is built from three requirements and nothing else: no edge changes length, no panel bends, and the panel’s corner angle is unchanged. Writing those out for a vertex grid gives three equations in four unknowns — one degree of freedom, which is the theorem about the Miura arriving as arithmetic — and solving them gives the vertex positions at any point of the motion.

The generator then checks the solution rather than trusting it. Every edge length in the folded state is compared with the flat pattern’s, and if any differs by more than a part in a billion it throws. The reported error sits at the level of floating-point rounding.

Poisson’s ratio is then a finite difference on that motion, and the chart’s central claim — that the value is negative everywhere — is asserted before the curve is drawn.

The correction is worth recording. An earlier version of the Miura figures took two published in-plane dimensions and paired them so that one grew while the other shrank. That is a positive ratio, printed under a caption asserting a negative one, with the numbers visible on the page contradicting the sentence beside them. It was found by trying to compute the ratio and getting the wrong sign.

Tunable, which is the real claim #

The interesting engineering property is not the sign. It is that the sign and the magnitude are chosen at design time.

Change the panel’s slant and the whole curve moves: a shallow slant gives a ratio near zero at the flat end and a steep one gives a large negative value immediately. Nothing else changes — the same sheet, the same fold count, the same material — and the mechanical response is different.

That is what “metamaterial” means, and it is worth separating from the loose use of the word. A metamaterial is a structure whose bulk properties come from its geometry rather than from its constituents, so that the properties can be designed. A folded sheet qualifies exactly.

It also means the design space is enormous and mostly unexplored. There are as many folded metamaterials as there are periodic crease patterns, which is a set nobody has enumerated, and the mapping from pattern to properties is not known in general.

The bending sign is the other way round #

There is a second property of a Miura sheet that is genuinely surprising and that this site’s machinery cannot compute.

Bend a Miura-folded sheet rather than stretching it. It curves into a saddle: bend it one way along its length and it bends the opposite way across its width. That is a positive Poisson’s ratio in bending, and it is the opposite sign to the in-plane one.

For an ordinary plate the two signs agree. A material with a positive in-plane ratio bends into a saddle; one with a negative ratio bends into a dome. The Miura does the reverse of what its in-plane behaviour predicts, and Mark Schenk and Simon Guest identified this in 2013.

The reason it cannot be computed here is instructive. Bending a folded sheet requires the panels to deform, so the rigid model has nothing to say — a rigid Miura has one degree of freedom and bending is not it. Getting the bending response means modelling the panels as elastic plates and the creases as torsional springs, which is a finite-element calculation rather than a geometric one.

So this essay states the result and attributes it rather than drawing it, which is the honest treatment of a claim the site’s own machinery cannot check.

What a unit cell is #

The word “material” is doing something specific here and it is worth unpacking, because a folded sheet is visibly not a continuum.

A metamaterial’s properties are defined on a scale larger than its cells. A Miura fold’s unit cell is four panels; a sheet of a hundred cells behaves, on the scale of the whole sheet, like a surface with a Poisson’s ratio, and the cell structure is invisible at that scale in the same way that steel’s crystal grains are invisible.

That is the whole justification for the word. A single Miura cell is a mechanism; a hundred of them are a material with mechanical properties, and the properties are the cell’s kinematics averaged.

Where the analogy breaks is at the edges. An interior cell is constrained by neighbours on all sides; a boundary cell is not, and behaves differently. For a sheet of a hundred cells about a third are on the boundary, so a real specimen’s measured ratio approaches the model’s from below and reaches it only for large sheets.

The other break is that a folded sheet’s cells are all in the same state, because the sheet has one degree of freedom. A material’s grains are in whatever state the local stress puts them. That is why a folded sheet’s properties depend on a single global parameter and a real material’s do not.

Stiffness is the other half, and it is harder #

Poisson’s ratio is a kinematic quantity: it describes how the shape changes, and it can be computed from geometry alone. Stiffness cannot.

To say how much force a folded sheet takes to compress, the creases have to be given a torsional stiffness and the panels a bending stiffness, and the answer depends on both. A Miura sheet is very compliant along its one degree of freedom and very stiff against everything else, and the ratio between the two can be four or five orders of magnitude.

That anisotropy is the property engineering actually uses. A deployable wants to be soft in the direction it deploys and rigid in every other, and a folded sheet delivers exactly that without any material being anisotropic.

It also explains a fact about paper models that is otherwise puzzling. A Miura-folded sheet feels stiff in the hand, and it is — in five of its six degrees of freedom. The sixth is so much softer that the sheet appears to have only one way of moving, which is precisely the claim the kinematics makes and which is here a statement about relative stiffness rather than about geometry.

Where the model stops #

Rigid panels, and real panels are not. The whole motion assumes the faces never bend. Paper bends slightly, which is why a paper Miura is more forgiving than the model and why a sheet-metal one is less.

Zero thickness. The packed state in the model has all layers coincident. A real one is as thick as the layer count, and getting thickness round a corner is what a manufactured version has to solve.

Zero thickness at the packed end. The model’s fully closed state has every layer coincident, which is where the accommodation techniques become the binding constraint on a real sheet.

Infinite sheets. The ratio computed here is a property of the unit cell. A finite sheet has edges, the edge cells behave differently, and the measured ratio of a real specimen approaches the model’s value from below.

Quasi-static. Nothing here is about how fast the sheet moves or how much force it takes. Those need a stiffness model, and stiffness is where the material properties come back in.

Rigid-foldability is assumed. The kinematics presumes the pattern folds rigidly to begin with, which most patterns do not.

Nothing about failure. The kinematics has no yield, no tearing and no crease fatigue. A real folded sheet fails at its creases long before it fails anywhere else, and how many cycles it survives is a materials question this model cannot touch.

One pattern family. Everything above is Miura against accordion. The eggbox pattern has a positive in-plane ratio, and a sheet combining Miura and eggbox regions has a ratio that varies across the sheet — which is a further design freedom this essay does not draw.

The surprise: the sheet has properties the paper cannot have #

The deepest thing here is not any particular number. It is that a folded sheet can have mechanical behaviour that no material can have at all.

A ratio that varies by an order of magnitude with the loading state is not a material property; materials do not have those. A ratio whose bending and stretching signs disagree is not a plate property. A single sheet whose response is different in different regions is not a homogeneous anything.

The folded sheet escapes all of those constraints because its behaviour is kinematic rather than constitutive. It moves the way a mechanism moves, and a mechanism’s response depends on where it is in its travel. What looks like a material with strange properties is a machine with ordinary ones.

That reframing is the useful takeaway, and it is why the same mathematics that folds a paper crane deploys a solar array. It is also why a twist tessellation behaves quite differently despite being made the same way: it is a different mechanism, so it has different mechanics, and no property of the paper distinguishes them. Neither is about paper. Both are about a surface that has been given a mechanism.

Why the word is worth defending #

“Metamaterial” is a fashionable term and it gets attached to things that do not earn it, so it is worth saying what the test is.

The test is whether the property survives changing the material. If a folded sheet’s negative Poisson’s ratio is the same in paper, in Mylar and in steel foil — and it is, because the computation that produces it never mentions the material — then the property belongs to the structure. If it changed with the material it would be a material property with a structural correction, which is a different and much commoner thing.

A folded sheet passes cleanly. The kinematics is a statement about lengths and angles; nothing about stiffness, density or bonding enters it. The same crease pattern in any inextensible sheet gives the same curve.

What does depend on the material is everything about force: how stiff the sheet is, how much it takes to fold, how many cycles it survives. So the honest division is that folding sets the shape of the response and the material sets its scale, and calling the result a metamaterial is a claim about the first half only.

Who found it, and when #

The mechanics arrived long after the pattern.

The pattern itself is much older than its mechanics; a single flat-foldable vertex tiled across a sheet is the whole of its geometry, and that was understood as folding long before anybody measured it as a material. Koryo Miura published the fold in 1970 as a packaging solution for large membranes in space, and the negative Poisson’s ratio is implicit in his analysis without being the point of it. It flew on the Space Flyer Unit in 1995.

The auxetic reading came from the materials community. Roderic Lakes made the first deliberately auxetic foam in 1987 and the term entered use shortly after; folded sheets were recognised as belonging to the same family through the 2000s.

Mark Schenk and Simon Guest’s 2013 paper is the one that treats a Miura sheet as a metamaterial properly, with both the in-plane and the bending behaviour, and the sign reversal between them. The design-space question — which patterns give which properties — is active and open.

The ladder from here #

Later rungs against this anchor: the eggbox pattern and its positive ratio. Combined sheets, with regions of each. Stiffness rather than kinematics, which needs a spring model at every crease. Bistable patterns and switching. The bending response and the sign reversal, computed. Curved-crease metamaterials, which are a different family again. And the inverse problem — given a desired response, find the crease pattern — which is the question the field would most like answered and has barely started on.

The properties are real, measurable and useful, and none of them is a property of paper.