A property you can dial

A material’s Poisson’s ratio is a number in a table. Cork is about zero, steel about a third, rubber close to a half. Look up the number, use it, and it does not change while the part is in service.

The map above is what happens when the same question is asked of a folded sheet. There is no number to look up. There is a surface, its values run over more than a factor of ten, and moving along it requires nothing but folding the sheet a little further — which is something the sheet does anyway, in service, as part of its job.

What is being measured #

Poisson’s ratio is the ratio of transverse contraction to axial extension: pull on something and see how much narrower it gets. Formally,

$$\nu = -\frac{\varepsilon_{\text{transverse}}}{\varepsilon_{\text{axial}}},$$

with the minus sign inserted so that ordinary materials, which do get narrower, have a positive value.

A Miura fold gets wider, and it does so because one vertex repeated leaves the sheet a single freedom, so its value is negative — the property that names it auxetic and the one it is famous for. That fact is a single sign and it is where most accounts stop.

What the map adds is magnitude. Each cell is computed by taking the solved rigid folding at two nearby points of the motion, measuring the cell’s two in-plane dimensions at each, and dividing the logarithmic changes. No closed form is quoted anywhere; the folding is solved from the requirement that no edge changes length and no panel bends, and the ratio is read off it.

The two axes #

The panel slant is a design choice, made once, before anything is manufactured. A shallow slant gives a sheet close to an accordion; a strong one gives sharply zig-zagged rows.

How far the sheet is folded is not a design choice. It is the state the sheet is in at a given moment, and a deployable spends its life traversing it: packed for launch, opening in transit, flat in service.

So one axis of the map is chosen at the factory and the other is chosen by circumstance, and the ratio depends on both. A design specified by its Poisson’s ratio has been specified at one point of its own motion, and the specification is silent about every other point.

Reading the map #

Three things are worth reading off it directly, because they are the sort of thing a chart makes obvious and a formula does not.

The steep corner. The most negative values sit where the slant is strongest and the sheet is barely folded. That is the corner where a small change in fold state produces a large change in ratio, and it is the corner a designer wanting a stable property must avoid.

The flat edge. The shallowest slants give values close to a half in magnitude and change slowly across the whole travel. They are the closest this family comes to a material, and the price is that the auxetic effect is modest.

The absent zero. Nothing on the map crosses into positive territory, and nothing reaches zero. The Miura family is auxetic everywhere it exists, which is stronger than the usual statement and is the sort of claim a grid can support and a single curve cannot.

What “material” is doing in “metamaterial” #

The word is a claim, and the map is a good place to examine it.

A material property is a number belonging to a substance, measured on a sample and transferred to a part made of that substance. Its two important features are that it does not depend on the shape of the sample and does not depend on how much the sample has been loaded — the first makes it tabulatable, the second makes it usable.

A folded sheet’s ratio fails both. It depends on the panel shape entirely, since that is where it comes from, and it depends on the configuration, which is what the map’s second axis is. Calling it a material property is an abuse of language that has been useful enough to survive.

The honest description is that a folded sheet is a mechanism whose kinematics happen to be describable in the vocabulary of continuum mechanics. That is a stronger position rather than a weaker one: mechanisms can be designed, and material properties can only be selected from a list.

The map has a mirror image, and their product is one #

There is a second map the figure does not draw, and it is not independent of the first — it is the first turned over, exactly.

The ratio is defined with one dimension on top and the other underneath. Measure it the other way round and the two logarithmic strains swap places, so

$$\nu_{wl} \cdot \nu_{lw} = 1$$

at every point of the design space, for the trivial reason that a one-degree-of-freedom sheet has both dimensions as functions of one parameter and the two ratios are reciprocals of each other by construction.

So the map above is one of two, and the other is its pointwise reciprocal. A cell reading $-0.5$ in one direction reads $-2$ in the other. A cell reading $-3$ reads $-\tfrac13$.

That is not a technicality about conventions. It says a Miura has one independent Poisson’s ratio rather than two, which is what having a single freedom means, and it says the number quoted for any folded sheet is meaningless without saying which pair of dimensions it was measured across.

Which forbids something the map appears to offer #

The design reading changes accordingly, and it changes in the direction of a constraint.

The map’s shallow-slant edge sits near a half in magnitude and is described as the closest this family comes to a material. Measured the other way that same edge sits near two, and two is not a modest number — it is more auxetic than anything in a materials table.

So there is no configuration in which a Miura is mildly auxetic in both directions. One of the two always has magnitude at least one, because the product is exactly one and both are negative. A designer who reports a gentle ratio has reported the gentler of a reciprocal pair and left the other unmentioned.

The line where the direction stops mattering #

The same identity supplies a landmark the map does not label, and it is the one point where the ambiguity vanishes.

A number equal to its own reciprocal is $\pm 1$, and these are all negative, so $\nu = -1$ is the only self-consistent value. On that curve the sheet contracts in both directions at the same logarithmic rate, and it is the only place where “the Poisson’s ratio of this sheet” is a well-posed phrase.

The map spans values from about a half to several, so it crosses that curve, and the crossing is a continuous line through the design space rather than a point. That line is the ridge a designer wanting a directionally neutral sheet should be reading off, and it is computable from the same solved folding the map already runs — no new machinery, one contour, and the one contour on the surface that means something on its own.

The tunable part, and what it costs #

The engineering reading of the map is that the ratio is a control input.

That has been exploited. A structure whose Poisson’s ratio changes as it deploys changes its stiffness distribution as it deploys, which can be used to make something compliant while packing and rigid when open. It is also a nuisance: a designer who wants a constant ratio across the working range has to choose a slant whose row on the map is as flat as possible, and the flat rows are the ones with small magnitude.

That is the trade the map makes visible and a single number cannot. Large auxetic effect and constant auxetic effect are incompatible in this family, because the same steepness produces both.

What a designer chooses, in order #

The map suggests that a designer picks a slant and lives with the row. In practice the choice is made in a different order, and the order is worth setting out because it explains why the ratio is rarely the thing being specified.

First comes the packed size, which fixes how far the sheet must close. Then comes the deployed size, which with the packed size fixes the number of cells. Only then is there a slant to choose, and by that point the slant is constrained by the thickness the panels will have — a strong slant makes long thin facets that stack badly and jam early.

So Poisson’s ratio is generally an outcome rather than an input: the designer discovers which row of the map the design landed on. That is not a criticism of the map — it is the reason to have one, in the same way that the packing figure is worth having next to the sentence it supports, since a designer who can see the whole surface can tell whether the row they landed on is a steep one, and can trade a little packing for a lot of stability if it is.

The other sign #

There is a folded surface with a positive ratio, and it belongs in this essay as a control case even though it cannot be drawn here.

The eggbox is a corrugation of alternating peaks and troughs, and stretched in one direction it grows in the other’s opposite sense: a positive Poisson’s ratio, from a pattern that looks superficially like the Miura’s cousin. Schenk and Guest studied the two side by side and the comparison is the standard reference for the claim that the sign belongs to the pattern.

The eggbox is not drawn here, and the reason is a rule this site keeps: it is not a folded flat sheet. Its surface cannot be developed from a plane without stretching, so it is a shell that has been formed rather than a sheet that has been creased, and every crease pattern on this site is one a reader could cut out and fold. Drawing an eggbox as a crease pattern would be drawing something that does not exist.

What survives from the comparison is the useful half: two corrugated surfaces of similar appearance, opposite signs, and no material difference between them. The sign is geometry.

Where the property comes from #

It is worth tracing the ratio back to the vertex, because the trace is short and it explains why the map has the shape it has.

A degree-four vertex has one degree of freedom. Repeating it periodically gives a sheet that also has one, because each vertex’s freedom is consumed by its neighbour’s choice rather than added to it. So the whole sheet’s configuration is a single number, and every dimension of the sheet is a function of that number.

Poisson’s ratio is then a ratio of two derivatives of those functions. It is negative because both dimensions fall together as the sheet closes, and it varies because the two functions have different shapes — one falls nearly linearly and the other with increasing steepness.

That is the sense in which the property is local: it is a fact about one vertex, propagated. Changing the panel slant changes the vertex, which changes the two functions, which moves the whole row of the map. Nothing about the sheet’s size or boundary enters.

Which theorem was checked, and how #

The figure asserts two things before drawing, and each is the negation of a way it could be quietly wrong.

Every value on the map must be negative. A single positive cell would mean the pattern is not doing what the caption says at that point of the design space, and a map with one such cell in a corner is exactly the sort of thing nobody notices. This check exists because an earlier figure on this site printed a positive ratio under a caption asserting a negative one, having taken two published dimensions and paired them the wrong way round; nothing caught it, because no gate compared a figure’s numbers with its sentence.

And the map must not be flat: the largest and smallest values must differ by at least a tenth of their magnitude. If they did not, the ratio would be a constant of the pattern after all and this essay would be arguing for something that is not there.

Underneath both is the isometry check inside the folding itself. Every state on the map is solved rather than posed, and the solver refuses any configuration in which an edge has changed length by more than a part in $10^9$.

What the picture cannot show #

The map is a grid of samples and the underlying surface is smooth. Fifty-six cells is enough to see the shape and not enough to locate a maximum, and no contour is drawn because a contour through this few samples would be an interpolation presented as a measurement.

More importantly, the map is of a rigid folding. Every panel is flat and every fold is a perfect hinge, so the sheet has exactly one degree of freedom and the ratio is a property of that one path. A real sheet’s panels bend a little — which is most of what separates paper from panels — and that adds compliance the map does not contain, and near the flat state the real behaviour departs from it substantially — the mechanism is at a singular configuration there and the panel bending dominates.

The map is therefore accurate away from its edges and least accurate at the left-hand column, which is the state a deployed array actually sits in.

The idealisation underneath #

Rigid panels, zero thickness, perfect hinges — all the things a panel is not — plus one more that is specific to this figure.

The ratio is computed on a single cell, using the cell’s two in-plane dimensions. That treats the sheet as infinite and periodic. A real Miura array is a finite piece with a boundary, the boundary rows are not constrained on both sides, and their behaviour differs from the interior’s by an amount that matters when the array is a few cells across. The map is the bulk value, and small arrays do not have bulk values.

The surprising connection #

The map is a design chart, and design charts of exactly this shape are how engineering handled material selection before there were databases: two axes of what can be chosen, contours of what results, and a working point read off by eye.

What is different here is that one axis is not a choice at all. Ashby charts plot property against property for substances that sit still; this one plots a property against the configuration of the part, which is a variable no material chart has ever needed.

That difference is the whole of what folded metamaterials add to the vocabulary. A conventional part is made of a substance with fixed properties and gets its behaviour from its shape. A folded part gets its behaviour from a shape that changes while it works, and the property follows the shape. Nothing about that requires paper, or even folding — the same argument reaches hardware that was never a sheet — it requires a mechanism with a small number of degrees of freedom and a geometry that can be computed, which is what a crease pattern supplies.

Who found it, and when #

Koryo Miura published the fold in 1970 as a packaging method for large membranes, and its negative Poisson’s ratio was noticed later; the fold predates the vocabulary that now describes it, in the same way that the Yoshimura pattern predates anybody’s interest in why it is there.

The systematic study of folded sheets as materials belongs to the 2010s. Mark Schenk and Simon Guest’s Geometry of Miura-folded metamaterials (2013) is the paper that put the Miura and the eggbox side by side and established the sign as a property of the pattern. Zhiyan Wei, Mahadevan and others gave the elastic theory shortly afterwards, and the configuration-dependence — the fact that the map has a second axis at all — was the point on which the mechanics literature and the origami literature converged.

The ladder from here #

This rung establishes that the property is a surface rather than a number. The rung above asks what else on the surface can be dialled: stiffness has the same character, and so does the packing ratio, and the three are not independent — they are three derivatives of one motion, which means a designer choosing one has chosen the others.

Below it, a material made of creases establishes that a folded sheet has properties the paper did not, which is the claim this rung refines into a measurement.