Nothing to average over

A material made of creases measured a folded sheet’s Poisson’s ratio off a solved rigid folding rather than quoting it, and found an accordion sitting at exactly zero. A property that can be dialled measured the Miura’s across its whole design space and made the point that a configuration-dependent property is not a material property — the number moves as the sheet folds, so it describes a position rather than a substance.

Both were measured on a sheet that repeats one cell, and that is the assumption this rung removes.

What a cell is, and why a panel is not one #

A ratio of this kind compares two dimensional changes: stretch the sheet one way and see what happens the other. To measure it locally, something local has to have a width and a height that change.

A panel will not do — a rigid folding’s panels are rigid by definition, which is the whole content of the word. A panel of a rigid folding is rigid, so nothing about it changes as the sheet folds, and measuring one reports zero for every question. The smallest thing that changes is a block of two panels by two — the unit that repeats in a Miura — and its width and height are the distances between the midpoints of its opposite sides. Those are what a cell contributes to the sheet’s overall size, and what a shrink factor is a ratio of; a corner-to-corner distance would measure a diagonal, which does not shrink the way the sheet does.

The ratio itself is a finite difference on the placement: change the fold angle a hundredth of a radian either way, place the sheet twice, and divide the relative change in width by the relative change in height. Nothing about it is quoted and nothing is assumed about the vertex.

The control, which has to come out flat #

On a Miura the nine cells report −0.15109, all of them, agreeing to 2 × 10⁻¹³. The sheet measured as a single cell spanning the whole mesh reports −0.15109 as well.

That is what it means for a folded corrugation to have a ratio. Every part of it does the same thing, so a number measured on any part is a number about the whole, and the object behaves like a material in the one respect that matters for the word: it is describable without saying where.

The agreement is also the check that makes the rest evidence. Nine cells measured independently, by a placement that walks the sheet vertex by vertex and does nothing to enforce uniformity, landing on one number to thirteen decimal places — a discrepancy there would mean the measurement was reporting the walk rather than the sheet.

And the sheet that has no cell #

A mesh with no two vertices alike still folds, still has a width and a height, and still has a well-defined ratio for the sheet as a whole: −0.175.

Its cells report −0.149, −0.003, −0.118, −3.532, −0.465, −2.692, −0.090, −0.298 and +0.438.

The spread is 3.97, which is twenty-two times the sheet’s own figure. One cell is essentially inert; one contracts more than three times as fast as it is stretched; and one goes the other way, narrowing where the sheet is widening.

The sheet’s number lies inside the range and equals none of the nine. It is an average, in the plain sense that it is what measuring the whole thing returns, and there is no part of the sheet it describes.

Why that is more than an inconvenience #

The word metamaterial carries a specific promise: that a structure made of parts can be treated as a material with properties, so that a designer specifies a ratio and a stiffness and hands them to somebody who builds a sheet with them. The promise rests on a step that is usually invisible — homogenisation, the replacement of a repeating structure by the average of its cell.

That step needs a cell. It needs the cell to be the same everywhere, and it needs the sheet to be large enough that the average is what a user experiences rather than the individual cells.

A mesh with no repeating cell fails the first requirement outright. Its whole-sheet number exists, is measurable and is reproducible, and it is not a property in the sense the word is being used: it does not predict what happens at any point, and a smaller piece cut from the same sheet would report a different one.

Averaging the cells gives the wrong answer by a factor of four #

“It is an average, in the plain sense” understates the failure, and the nine numbers say so.

Add them and divide by nine: the cells’ arithmetic mean is −0.768, against the sheet’s −0.175. Not close, and not off by a rounding — off by four and a half times, and in the direction of a sheet that contracts far harder than the real one does.

The median is −0.149, within fifteen per cent of the sheet’s figure. So the distribution is not merely spread; it is heavy-tailed, with two cells at −3.53 and −2.69 dragging the mean away from where the sheet actually is.

Which is a fact about ratios, not about this mesh #

The reason is structural and worth naming, because it applies to every attempt to homogenise a quantity of this kind.

A Poisson’s ratio is a quotient of two small changes. A cell whose height barely moves has a near-zero denominator, and a near-zero denominator makes the quotient arbitrarily large without the cell doing anything remarkable — the cell at −3.53 is not contracting violently, it is a cell whose stretch direction hardly moved.

The sheet’s number is not exposed to that, because it is a quotient of totals: the whole sheet’s width change over the whole sheet’s height change. Small denominators are summed away before the division happens.

So the ratio of averages and the average of ratios are different numbers, and here they differ by 4.4×. That is the exact sense in which there is nothing to average over. It is not that the average is unrepresentative of a spread — it is that the operation homogenisation performs is not the operation the sheet performs, and only a repeating cell makes the two agree.

The Miura is not exempt either #

There is a second reading of the same measurement, and it strengthens the earlier rung rather than qualifying it.

The Miura’s number is constant across the sheet and it is not constant through the motion. Measured early in the fold and late in it, the ratio is a different number, and that was the previous rung’s finding: a configuration-dependent property is a property of a position, not of a substance.

So there are two separate ways for the word “property” to fail. It fails across space on a sheet with no repeating cell, and it fails across configuration on every folded sheet including the Miura. A material’s Poisson’s ratio does neither.

That widening is worth its own sentence. The Miura’s width rises to 1.039 of the flat sheet’s part way through the fold and then falls to 0.659 — so a package sized for the flat sheet and for the folded stack is not big enough for the thing in between.

What the outlying cells are doing #

The nine numbers are not noise around a mean and it is worth saying what the extremes correspond to, because that is what turns a spread into a mechanism.

The cell reporting −3.532 is the one whose height barely changes as the sheet folds while its width changes a great deal. Its panels are arranged so that most of the fold angle is spent turning them about a nearly horizontal axis, so the block collapses sideways and keeps its length; the ratio is large because its denominator is small.

The cell reporting +0.438 is the opposite arrangement, and the sign is the interesting part. As the sheet is folded and every cell’s height falls, that one’s width rises. It is being stretched by its neighbours rather than driven by its own geometry, which is a thing a cell of a repeating tessellation never does because its neighbours are doing exactly what it is.

The cell reporting −0.003 is inert: it changes shape hardly at all. On a sheet whose whole purpose is to fold, one ninth of it is going along for the ride.

None of that is visible in the whole-sheet number, and none of it would be recoverable from it. A sheet described by −0.175 is described as though it were nine copies of a mild contraction, and it is one collapse, one inversion, one bystander and six intermediates.

Which theorem was checked, and how #

The measurement rests on the placement, and the placement is the machinery built for the joint solve to put a folded mesh where its fold angles say it is. It is checked first and hardest. Every edge and both diagonals of every panel keep the lengths the flat pattern gives them; the four copies of every vertex, placed by a walk that arrives from different directions and does nothing to make them agree, land on one point to a part in a million million; and the dihedral angle read back out of the geometry is the fold angle that went in.

A cell is measured only on a placement whose vertices close. A ratio computed from panels that do not meet is a number about nothing, and the check requires the gap to be zero before any ratio is believed.

The two claims are then separated on purpose. The Miura’s cells must agree — a spread there would say the measurement is noisy. The general mesh’s cells must disagree, and by more than a rounding: the check requires a spread above 0.05 and requires the cells to differ in sign, because a spread of amounts is a weaker claim than a disagreement about direction.

The two ways a number stops being a property #

Setting the failures side by side makes the argument compact, and each of them has a different remedy.

Failure across configuration. Every folded sheet’s ratio depends on how folded it is, because the mechanism’s two dimensions are not proportional through its motion. The remedy is to quote the number with a configuration attached, which the earlier rung recommended and which is what a careful account of a folded metamaterial already does.

Failure across space. A sheet with no repeating cell has a different ratio at every place. There is no remedy — quoting a distribution instead of a number is a description of the sheet, not a property of a material, and any smaller piece cut from it reports something else.

The Miura fails the first and passes the second. The general mesh fails both. And there is no sheet that passes the first, because the first failure is a consequence of being a mechanism at all: a material’s ratio does not depend on how much it has already been deformed, and a mechanism’s does.

Where the model stops #

Everything here is geometry. There is no stiffness, no stress and no energy anywhere in the measurement, and the ratio computed is a ratio of dimensional changes along a mechanism’s path — which is what a folded sheet’s Poisson’s ratio is usually taken to mean, and is not what an elastic modulus is. A sheet whose panels can bend behaves differently and that is a materials question with a material in it.

The sheet is also small. Nine cells is few enough that “the average describes no cell” is unsurprising, and a large disordered corrugation might well have a distribution narrow enough for an average to be useful. The sheet also has a tolerance problem that would bite long before it was large enough to test, which is a reason the question has not been asked of a real object. What this rung establishes is that the justification for using one has gone, not that the number is always useless.

What the picture cannot show #

The cell figure draws nine numbers in nine boxes and the boxes are the flat pattern’s cells rather than the folded sheet’s. There is no good way to show a ratio in place on a folded object: the quantity is a rate of change, and a still picture of a folded sheet has no rate in it.

Nor can a picture show that two of the cells disagree about sign in a way a reader can feel. A cell whose width rises while its neighbours’ fall is doing something a folded sheet is not supposed to do, and it is drawn as a number in a box like the rest. Nothing shows the averaging. The sheet’s own figure is computed by measuring the whole mesh as one cell, which is a different measurement from taking the mean of the nine — and on this mesh the two are close but not equal, because the cells are of different sizes and the whole-sheet measure weights them by how much of the sheet they are.

What a designer can still do with it #

The result is negative and it does not leave a designer with nothing, so it is worth setting out what survives.

The whole-sheet number is still a measurement. It is reproducible, it is computed from the folding rather than fitted, and it predicts what the sheet as a whole does when the sheet as a whole is folded. What it does not do is predict what a part does, or what a differently-sized piece of the same design would do.

The cell map is available. Every number in this essay came from placing the folded sheet and measuring blocks of it, and a designer with a candidate mesh can have the map rather than the average — which is more information and not less, and is the honest object to hand to somebody deciding where to attach something.

And the family is large. Sixteen free directions is a great deal of room, and there is no reason the spread of the cell ratios should be the same everywhere in it. A search for the member whose cells agree most closely is a well-posed question with a computable objective, and its answer would be a sheet with no repeating unit that can nevertheless be described by one number.

The generalisation #

The useful form of this is about when averaging is allowed rather than about origami. A property of a structure is a property of a material when the structure has a repeating unit, the unit is small compared with the thing being made, and the response is the unit’s response. Take away the repeating unit and the average survives as an arithmetic operation and stops being a prediction.

Folding gives an unusually sharp example because the two cases are so close together. The Miura and the solved mesh are both developable quadrilateral meshes, both flat-foldable at every vertex, both rigidly foldable with one degree of freedom, and both look like the same kind of object drawn slightly differently. One of them has a Poisson’s ratio and the other has a number.

The design consequence is the one worth carrying away. The general meshes the joint solve produces are a sixteen-dimensional family and they are far more numerous than the repeating ones — so almost every rigidly folding quadrilateral mesh is a sheet whose behaviour cannot be summarised. The ones that can be summarised are the thin, special, repetitive subset the field has been working in.

Who found it, and when #

Homogenisation as a technique is old and its conditions are well understood wherever it is used carefully: a repeating cell, a separation of scales, a response that is the cell’s. Folded metamaterials are a young enough subject that the conditions are usually satisfied by construction — the patterns studied are tessellations, so they repeat by definition — and the question of what happens without a cell has not needed asking.

It becomes askable only once non-repeating meshes that fold exist to ask it of, which is a thing the joint solve produced. The measurement itself is then immediate, which is a reasonable summary of why it has not been made.

Where the ladder goes next #

The obvious continuation is the distribution rather than the range. Nine cells give a spread; a larger solved mesh would give a histogram, and whether the cells’ ratios cluster or scatter decides whether a disordered folded sheet is a material with a wide tolerance or a collection of unrelated parts.

The other direction is to ask which member of the sixteen-dimensional family has the narrowest spread. If the solutions include some whose cells nearly agree, they would be sheets with no repeating unit that can still be described by one number — which is what a designer wanting both freedom and predictability would ask for.