The property a patch does not have

The claim that makes a corrugation interesting to anybody outside this subject is that it stops being a sheet with creases in it and starts being a material. A Miura has a Poisson’s ratio, a packing behaviour and a stiffness that the paper never had, and those properties belong to the pattern rather than to the paper — which is why the same folds work in steel, in a solar array and in a stent.

Every one of those words is a limit. A Poisson’s ratio is a ratio of strains in a body large enough that its surfaces do not matter. A packing ratio is a ratio of volumes in a medium with no edge. When they are quoted for a folded sheet, they are quoted for something that is a patch of a few units across with a boundary running all the way round it — and the boundary of such a patch is a third of it at any size a page or a sheet of paper reaches.

What the measurement is #

The compaction of a folded sheet is the sheet’s area divided by the footprint it packs into, which is the same thing as the average number of layers over that footprint. It is read off the placed folded state rather than predicted: the panels are put in the plane by composing reflections, a fine grid is laid over the result, and the layers over each cell are counted.

That makes it a genuinely different quantity from the shrink the construction reports, which is an area accounting done on the flat sheet — how much paper the pleats take up, computed before anything is folded. The construction’s number for these patches runs between 1.2 and 2.4. The measured compaction runs between 3.6 and 5.0. They are not versions of each other and they do not disagree; they answer two questions, and only the second is about the folded object a reader is holding.

The number moves with the size of the unit #

Fold the same tiling on the same square at six unit sizes and the compaction is not one number with noise on it. It is a rising sequence.

On the square grid: 3.89 layers when the twists are half the sheet across, 3.64, 4.04, 4.30, 4.34, and 4.79 when they are under a fifth. On the triangular grid: 3.92 rising to 4.80. On the honeycomb: 4.05 to 4.88. Three tilings, three different constructions, the same shape of answer — a rise of about a quarter across the range, monotone but for the wobble that comes from a lattice landing differently on a square, and not settled at the fine end.

The reason is not subtle once the two columns are read together. The share of units on the paper that the rim cuts falls from about nine tenths to about four tenths across the same range. A cut unit is a unit that does not pack: it has lost some of its pleats to the edge, its paper stays flatter, and it contributes to the footprint without contributing its share of layers.

What is held fixed, and why it has to be #

The experiment is one tiling, one square, one twist angle, and six unit sizes. Everything else is deliberately not varied, because almost everything else moves this number.

The twist angle stays at 0.35 radians throughout. It sets how much paper each pleat takes and therefore how deep the folded stack is; sweeping it as well would produce a surface rather than a sequence and would confound the two effects.

The sheet stays a unit square. Changing the sheet changes the perimeter-to-area ratio, which is the very quantity under test, so it has to be the constant and the unit size has to be the variable.

The pleats keep the same share of the room available between neighbouring twists. The construction fixes the polygon sizes up to one common factor, and that factor is set by how much of the gap between two twists the pleats use. Holding it fixed is what makes a small unit a scaled copy of a large one rather than a differently proportioned pattern.

What is left free is the one thing being measured: how many units land on the paper and how many of them the paper’s edge cuts. That is why the sequence can be read as a boundary effect rather than as six unrelated patterns — the only difference between the first row and the last is a magnification.

Some of these numbers converge and some do not #

The twist tessellations settle towards about five layers. A Miura does not settle at all, and the difference is worth more than either number.

Folded at increasing sizes, a Miura’s compaction runs 3.03, 5.48, 8.14, 9.18, 13.77 and 14.76 layers, for patches from two cells across to eight by six. That is not converging on anything: it is roughly proportional to the number of rows, because a Miura closes into a slab about one cell in footprint with every row stacked on top of the last. Double the paper in that direction and the layer count doubles.

So compaction is an intensive property of a twist tessellation and an extensive property of a Miura. For the twist, “about five layers” is a fact about the pattern that a large enough patch would reproduce. For the Miura, the same phrase is a fact about the piece of paper, and quoting it without the size is quoting nothing.

That distinction is not visible in a single figure of a single patch, and it is invisible in the way both are usually reported: one pattern, one number, no size. The two behaviours look identical until the paper changes.

The rim does not pack #

Why a cut unit packs less is visible one panel at a time, and this collection has already measured it from the other direction: a panel carrying a raw edge of the paper lies over fewer of the other panels than a panel that does not — 18.0 against 21.0 on a Miura, 31.0 against 37.7 on a waterbomb, and never once the other way round on any pattern with both kinds.

That is the whole mechanism. Layers accumulate where panels are stacked on panels, and a panel at the edge has neighbours on fewer sides. The effect is not a small correction at the margin; it is a deficit on every panel in a band one unit deep, and the band is most of a small patch.

The same logic runs through every bulk property, not just this one. A Poisson’s ratio is measured by comparing two dimensions of the folded object as it opens, and the outermost row of cells is the row whose motion is least constrained — it has no neighbour to be pulled by on one side. A stiffness measured across a patch is measured across a boundary as much as a bulk. In each case the patch reports the material’s behaviour diluted by however much of it is edge.

Two ways a patch fails to be a material #

The boundary is one of them and this collection has already found the other, which is worth putting beside it because the two are independent.

On a sheet with no repeating cell, the Poisson’s ratio measured cell by cell runs from −3.5 to +0.4 — some cells widening while others narrow — and the sheet’s own overall figure describes none of them. That is a failure of homogeneity: there is no single cell whose number is the sheet’s number, so averaging produces a figure nothing in the sheet has.

The boundary is a failure of extent: even where every cell is identical and the average is meaningful, a finite piece of the pattern has a rim whose cells behave differently because of where they are rather than because of what they are.

A Miura patch has the second problem and not the first; a quadrilateral mesh has both. The distinction matters for what to do about each. Inhomogeneity is repaired by measuring cell by cell and reporting the spread, which is a change to what is reported. Finiteness is not repaired by any change to the reporting — the number would have to be measured on a bigger patch, and there is no bigger patch, because the paper is the size it is.

Extrapolating, carefully #

The obvious response is to fit the trend and quote the limit. That is defensible and it should be done with its assumption showing.

The share cut falls like the perimeter over the area — about four unit widths over the sheet’s side — so the deficit should fall the same way, and the compaction should approach its limit like one over the number of units across. Fitting that shape to the square grid’s six measurements puts the limit somewhere near five layers, against 4.79 at the finest size drawn. The honeycomb and the triangular grid land in the same place.

Two cautions come with it. The first is that six points spanning a factor of three in unit size, on a sequence that wobbles by a tenth of a layer as the lattice lands, do not pin an asymptote to better than about ten per cent. The second is that the extrapolation is a statement about a pattern with no edge, and nothing in this collection has ever folded one — the unit is verified and the plane is not, and a limit inferred from patches is inferred from exactly the objects whose finiteness is the problem.

What the deficit is worth, per unit #

The size of the effect can be put in one number, and it is larger than the differences between patterns that get compared.

Across the square grid’s six sizes the compaction rises by 0.9 layers while the cut share falls by about half. Reading those together, a unit cut by the rim is worth roughly two layers less than a whole one — a cut unit contributing about three where a whole one contributes about five. On a patch where nine tenths of the units are cut, the average is nearly the cut number; on one where two fifths are, it is most of the way to the whole number.

That also says how big a patch would have to be for the number to be worth quoting to a tenth of a layer. The deficit is the cut share times two layers, so a tenth of a layer needs a cut share of five per cent, which needs about eighty units across. On a 150 mm sheet that is a unit under two millimetres, with pleats a fifth of a millimetre.

The comparison worth making is with what separates the patterns themselves. The three tilings measured here end within a tenth of a layer of each other at the fine end and are half a layer apart at the coarse end — so at the sizes actually drawn, which tiling it is matters less than how much of it is edge. A figure comparing two tessellations at different unit sizes is mostly comparing their boundaries.

The two extrapolations do not agree #

The essay reaches its limit twice by different routes and the two answers differ by more than the caution attached to either.

The first route fits the shape of the boundary effect and lands near five layers. The second computes a deficit of about two layers per cut unit, which is a stronger statement and can be run backwards.

Take the endpoints. At a cut share of 0.9 the compaction is 3.89, so the limit is $3.89 + 2 \times 0.9 = 5.69$. At a cut share of 0.4 it is 4.79, giving $4.79 + 2 \times 0.4 = 5.59$.

Both give about 5.6, and the two agree with each other to within two per cent while disagreeing with the curve fit by twelve.

Which makes the finest patch further from the limit #

The difference matters for the essay’s own practical conclusion.

Against a limit of five, the finest patch drawn — 4.79 layers at units under a fifth of the sheet — is at ninety-six per cent, and the remaining boundary effect looks like a rounding correction.

Against 5.6 it is at eighty-six per cent, and the shortfall is 0.8 layers rather than 0.2. That is four times the gap, and it is larger than the whole spread between the three tilings measured.

So the finest patch this collection can draw still understates the pattern’s compaction by about one layer in seven, which is the essay’s argument arriving with a bigger number than the essay gives it.

The reason to prefer the deficit route is that it is built from a mechanism rather than from a curve’s shape. Two layers per cut unit is a statement about what a rim panel does, corroborated independently by the rim lying over fewer panels than the interior on every pattern with both kinds. The one-over-units fit assumes the deficit falls like the perimeter and has six wobbling points to constrain it.

It also sharpens the size estimate at the end. Reaching a tenth of a layer needs the deficit down to that, which at two layers per cut unit needs a cut share of five per cent — the essay’s eighty units across, unchanged. What changes is where the number is heading: not a little above 4.8, but a good deal above it.

What a reader should do with a quoted ratio #

Three things follow for reading any figure that reports a bulk property of a folded pattern, including the ones in this collection.

Ask how many units across. The number is not comparable between a four-across patch and a nine-across one, and the difference between those two is larger than most of the differences between patterns that get compared.

Ask whether the rim was counted. Some measurements here deliberately exclude the cut units — the construction’s shrink is computed over whole units only, so that it describes the tessellation rather than the picture. That is the right choice for that number and it means the number is about a different region of the paper than the figure beside it shows.

And do not compare a patch’s number with a published bulk figure. Engineering literature on folded cores reports properties of cores many cells across, where the boundary share is a few per cent. A four-across paper patch is not a small version of that; it is a different regime, and the direction of the difference is known — the patch always understates the compaction.

Where the engineering puts it #

The same question has a settled answer outside paper, and it is settled in the direction this argues.

A folded core in a sandwich panel, a solar array, an airbag or a stent is many cells across in the direction that matters, and its designers do not quote a patch’s number — they quote a cell’s, derived from the unit and its neighbours, and treat the edge as a separate problem with its own name. The boundary of a deployable is where it attaches to something, where the seals go, and where the analysis stops being periodic; it is engineered rather than averaged.

That is the practice this measurement recommends for figures too. Report the unit’s behaviour from the unit, and report the patch’s behaviour as a patch’s — two numbers, not one, with the second carrying its size. The two agree only in a limit no sheet of paper reaches.

What is not being claimed #

Not that the properties are illusory. A Miura patch really does open in both directions at once, and its negative Poisson’s ratio is visible in four minutes of folding. What is finite is the number, not the behaviour.

Not that the trend is the whole story. The compaction also depends on the twist angle, on the tiling, and on how the lattice happens to sit on the square — the wobble in the sequence is that last effect, and at these sizes it is a tenth of a layer, which is a fifth of the whole rise being measured.

And not that a bigger patch is available. A twist tessellation twenty units across on a 150 mm sheet has pleats under a millimetre wide, which is past what paper will hold as two distinct creases and past what a page can draw. The regime where a patch is a material is one neither the reader nor the figure can reach, which is exactly why the patch’s number needs its size printed next to it.