A metamaterial with no edge

A mechanical metamaterial is a structure whose behaviour comes from its geometry rather than from what it is made of, and a folded sheet is one. Its properties — how it stretches, how it shears, how it stiffens — are quoted per unit cell, on the reasoning that the material repeats and one cell stands for the rest.

Every cell measured in this collection has been cut out of a patch. A cut cell has a rim, and a rim is precisely the place where a repeating material stops being like itself.

What a unit cell is supposed to be #

The idea is that a repeating structure is completely described by one period plus the rule that it repeats. Measure the period’s response and the whole material’s follows.

That works when the period being measured really is a period of the material — which means its opposite faces are joined to each other, because in the material they are joined to the neighbouring cells and the neighbouring cells are copies.

Cut the cell out instead and its edges become free. That is a different object with different degrees of freedom, and the difference is exactly the neighbours it no longer has.

The measurement #

On a two-period square twist cell: cut out of the plane, forty free letters over twenty-five panels. Glued into a torus, thirty-two letters over sixteen panels.

Eight of the forty letters — a fifth of them — belong to creases the rim divides, and a divided crease is two letters that need not agree with each other. In the material they do agree, because they are one crease.

So a cut cell has degrees of freedom the material does not have, and any per-cell property measured on it is measured on an object with too much freedom.

The rim, as a fifth of the freedom #

The number is worth dwelling on, since a fifth is larger than most corrections in this area.

A two-period square twist cell has forty free letters cut out and thirty-two glued. Eight letters — twenty per cent — are creases the rim divides, and each of them is a choice that exists only because the cell was cut.

At one period the fraction is worse: twelve letters cut out and eight glued, so a third of the freedom is rim. At three periods it is better: eighty-four and seventy-two, about a seventh.

That is the ordinary behaviour of a boundary — its share falls like one over the cell’s linear size — and it means the error in a per-cell measurement made on a cut cell is worst on the smallest cells, which are the ones anybody would use.

A materials person would recognise the shape immediately. It is the same reason a finite-element model of a periodic structure uses periodic conditions rather than a free cell: the free cell’s boundary contributes compliance the material does not have, and the contribution is proportionally largest on the smallest model.

What the rim contributes in kinematics #

Since the essay claims the kinematics is unaffected, it is worth being careful about that claim, because a rim contributes to the mechanics of a real cell even when it does not contribute to the mathematics here.

A physical cell of Miura cut from a sheet has edges that can flap. Its panels near the boundary are attached on fewer sides, so they move more freely, and a stiffness measured on such a cell is lower than the material’s.

That is a real effect, it is well known, and it is not what this essay is about. What this collection computes is combinatorial — which assignments of mountain and valley are consistent — and combinatorially the local conditions are unchanged by the sheet, because they are conditions at vertices and no vertex moved.

So there are two boundary effects and they should not be run together. The mechanical one is familiar and is handled by taking a big enough sample. The combinatorial one is the one measured here, and it is not fixed by taking a bigger sample, only reduced.

A tunability example #

The clearest place the correction bites is tunability, so it is worth working through.

A folded metamaterial is tunable when it has a family of configurations with different properties, and the family is chosen by which creases are mountains and which are valleys. So the number of available configurations is the number of consistent letterings, and the range of properties is what those letterings produce.

On a cut cell, the count includes letterings that differ only in the letters on divided creases — letters that, in the material, are the same letter as their partner on the far edge. Those are not distinct configurations of the material; they are distinct configurations of the sample.

How large that overcount is depends on the cell and it is not small: eight free letters on a two-period cell is up to two hundred and fifty-six times the count, before the vertex conditions cut it down.

The corrected count is the one on the glued cell, and it is available now. Nobody has computed the ratio, because the counts themselves are searches and the comparison would be a search apiece across a family of cells.

Where the honest boundary of this essay is #

Two limits, stated plainly.

Nothing here has been measured on an actual metamaterial property. The measurements are of free letters, panels, search cost and verdicts. Poisson’s ratio, stiffness and deployment force have not been computed on a glued cell, because this collection does not compute them at all.

The argument that kinematics survives is structural, not empirical. It rests on the vertex conditions being unchanged by a gluing, which is checked, and on the kinematics being a consequence of the vertex conditions, which is standard and is not checked here.

So what the essay establishes is that a cut cell is not the material’s cell, by how much in the quantities this collection measures, and which side of the line the usual metamaterial quantities fall on. It does not establish a corrected value for any metamaterial property, and it should not be read as claiming one.

Which properties are affected #

Not all of them, and the split follows a clean line.

Local properties are unaffected. How a single vertex moves, what the sector angles are, what the fold angle relationship is at a degree-four vertex: these are computed at a point and the sheet does not enter. The Miura’s single degree of freedom is a statement about a vertex propagated across the pattern, and it is the same statement on any sheet.

Global properties are affected. How many consistent letterings the cell admits, whether a periodic folded state exists, what the layer order is, how much a search costs.

The properties metamaterial work usually cares about — Poisson’s ratio, stiffness, the shape of the folding path — are in the first category, because they come from the kinematics of one vertex. So the correction is narrow.

Where it is not narrow #

Two places where the distinction bites, and both are about counting rather than about motion.

Counting configurations. How many distinct folded states a cell has is a global count and it depends on the sheet. A cut cell has more, because it has more letters to choose.

Tunability. A property one can dial is a property that varies over a family of configurations, and the family is what the count above counts. A cut cell’s family is larger than the material’s.

So a claim of the form this cell admits so many configurations is a claim about the object the cell was cut out of, and the material’s cell has fewer.

The other correction: the folded period #

There is a second and less obvious way a per-cell measurement can be wrong, and it applies to one of the families here.

A cell is a period of the drawing. It need not be a period of the folded state: the Yoshimura’s fold carries one drawn column onto the next by a turn of two hundred and forty degrees, so its folded state repeats every third column and its drawing repeats every one.

Any property of the folded object quoted per drawn cell is therefore quoted per third of a unit on that family.

That is a correction to the unit rather than to the object, and it is the more insidious of the two: a measurement on the wrong sheet is at least a measurement of something, and a measurement per wrong unit is a number with the wrong dimension attached.

What a materials person would do #

The distinction is standard elsewhere and it is worth borrowing the standard vocabulary.

Computing the properties of a periodic structure is normally done with periodic boundary conditions: the cell’s opposite faces are constrained to move together, so that the cell behaves as though it were surrounded by copies of itself. That is exactly the gluing above, and it is done as a matter of routine in the analysis of periodic solids.

The reason it has not been done here is that this collection’s cells are combinatorial rather than elastic. There is no stiffness matrix to constrain; there is a crease pattern, a set of letters, and a search. Applying periodic conditions to that means identifying creases and panels, which is the construction this phase is about and which did not exist before.

So the correction is: use periodic conditions, which for a folded metamaterial means glue the cell.

What is not being averaged over #

There is a companion result in this collection that sharpens the whole essay, and it is worth pointing at.

A patch has nothing to average over is the observation that a metamaterial’s properties are usually reported as averages over a cell, and that a cell is not a sample of anything — it is the whole of the structure, repeated, so an average over it is not a statistical quantity.

That is right and this essay adds the other half: the cell being averaged over is the wrong cell. It has a rim, the rim is not part of the material, and its share of the cell’s freedom is a fifth at two periods and a third at one.

Together the two say something quite specific about how a folded metamaterial should be measured. Not by sampling — there is nothing to sample. Not on a cut cell — that has freedoms the material lacks. On one cell of the material, with its edges identified to itself, which is one object, computed exactly, with no statistics anywhere.

That is the standard practice for periodic solids and it has not been the practice here, for the ordinary reason that the identification did not exist.

Why the word material does the damage #

A last observation about vocabulary, since the whole confusion is carried by one word.

A material is homogeneous by definition: the same everywhere, with properties that do not depend on where in it anybody looks. That is what licenses quoting a property per unit cell, and it is why the phrase unit cell carries so much weight.

A patch is a piece of something, with an edge, and the edge is where the homogeneity stops.

Calling a patch a material’s cell borrows the first word’s licence for the second word’s object. The borrowing is invisible because a patch of a repeating pattern looks homogeneous — it is the same drawing everywhere — and the inhomogeneity is not in the drawing. It is in which creases are the same crease, which is a fact about the sheet and not about the picture.

That is the same shape as everything else in this phase: the sheet is a parameter that no drawing records, and the properties that depend on it are exactly the ones nobody thought to attach it to.

The measurements that would follow #

If somebody wanted to do this properly, the list is short and none of it is hard.

Count the consistent letterings of a glued cell and of the corresponding cut cell, on the Miura at two and three periods. That is two searches apiece and it gives the overcount directly.

Do the same on a cylinder rather than a torus, which is the sheet a tube of the material actually has, and see whether the answer sits between.

And check the folded period of every pattern being used as a metamaterial, which is a handful of gluings and settles whether the unit is right.

None of that is done. The construction exists, the counts are affordable, and the reason it is recorded rather than run is that it is a different measurement from the one this phase was making.

Which sheet a material’s cell actually is #

There is a subtlety about which gluing corresponds to a real material, and it is worth getting right.

An infinite sheet of Miura, extending in both directions, has a cell whose four edges are all continuous with neighbours. The object matching that is the torus — both pairs identified — and that is the sheet with periodic conditions in both directions.

A tube of Miura, finite round and indefinite along, has cells that are continuous with neighbours in one direction and continuous with more of the same tube in the other. The object matching a cell of that is the cylinder, glued round.

A finite panel of Miura, a metre square, cut and installed — which is what most applications actually use — has a genuine boundary, and its cells near the edge really do have the extra freedom. For that object the cut cell is right at the edge and wrong in the middle.

So the correct sheet depends on what is being modelled, and the useful rule is: a cell in the interior of a material is a glued cell, and a cell at its boundary is a cut one. A measurement quoted as the material’s should use the first.

Most metamaterial work means the interior, since the whole point of a metamaterial is behaviour that scales, and behaviour at an edge does not.

The correction, stated for use #

Three sentences a reader could act on.

If the quantity is kinematic — how the structure moves, its degrees of freedom, its Poisson’s ratio, its folding path — the cut cell is fine, because those come from local conditions the sheet does not change.

If the quantity is a count — configurations, letterings, folded states, the size of a tunable family — the cut cell overcounts, by up to a factor of two per divided crease before the vertex conditions bite, and the corrected object is the glued cell.

If the pattern’s fold turns rather than slides — which is checkable and is true of the Yoshimura — then the unit itself is wrong by the folded period, and every per-cell quantity has to be rescaled.

The first covers most published work. The second and third cover the parts nobody has been computing, which is a reasonable position for a correction to be in.

What is worth taking #

The honest position is that most metamaterial measurements are unaffected and the reason is structural rather than lucky.

Metamaterial work is about kinematics — how the structure moves — and kinematics is local. The whole of the Miura’s behaviour as a material comes from a degree-four vertex with one degree of freedom, repeated, and repeating a local fact does not make it global.

What is affected is anything that counts. Configurations, letterings, folded states, search cost: each is a count over the whole cell, and each is measured on an object with a fifth more freedom than the material has.

And the unit itself is affected on any family whose folded period is not its drawn one, which is one family in ten measured and is a thing nobody has been recording.