Bought with holes

Pull on almost anything and it gets narrower. Pull on a Miura-folded sheet and it gets wider, and that single reversed sign is what the folded-sheet literature is largely about.

The measure of it is a ratio: how fast one dimension grows against how fast the other does, with a sign convention that makes ordinary behaviour positive. For a folded sheet the value is negative and it is also a moving target — it depends on the panel shape and on how far the sheet is folded, so a sheet specified by that number has been specified at one instant of its own motion.

There is a cut sheet that does the same thing and does not move.

The flat line is the finding. It is not put in anywhere; it is a finite difference of two dimensions measured off the drawn tiles, taken exactly the way the curve beside it is taken, and it is minus one to machine zero at every opening sampled.

The sheet, and the one equation it has #

The object is as simple as anything on this site. Take a square sheet, rule it into squares, and cut along every edge of every square except at the corners, where a point of paper is left joining each tile to its diagonal neighbours. Every tile is now attached to four others, each by a single point.

Turn one tile and its four neighbours must turn the other way, because the corner joining them is a hinge and the tiles are not allowed to change shape. That is the entire mechanism. It leaves exactly one equation to satisfy: the corner that tile A offers must be the same point as the corner tile B offers, and requiring that fixes how far apart the tile centres sit.

For tiles of side a turned through an angle θ, the lattice pitch is

$$P(\theta) = a\,(\cos\theta + \sin\theta),$$

and nothing else is free. At θ = 0 the tiles are edge to edge and the sheet is solid; at θ = 45° the pitch is a√2, the holes are squares the size of the tiles, and the motion is over — turning further would drive tile into tile.

The pitch is the same function in both directions, and that is the whole of the argument. Pull the sheet one way and the pitch grows; the pitch is what sets the spacing the other way too, so the other way grows by the identical factor. The ratio of the two logarithmic changes is therefore minus one, at every opening, without a kinematics being solved anywhere.

What opening it costs #

Nothing has been removed. Every cut is a slit, the tiles are the same tiles at every stage, and a slit that removes no paper changes no angle anywhere — the turn of paper at every point of every tile is still a full one. The sheet grows anyway, and every square millimetre of the growth is hole.

The bookkeeping is short. One repeating cell of the lattice has area P², it contains exactly one tile of area a², and the rest of it is hole. So the solid fraction is a²/P², and since P² = a²(1 + sin 2θ) the fraction is

$$\frac{1}{1 + \sin 2\theta}.$$

That runs from 1 at the closed position through two thirds at fifteen degrees — exactly two thirds, since sin 30° is a half — to one half at forty-five, exactly. Fully open, half of every cell is hole.

It is worth pausing on how cheaply that arrives. The whole of it is one shoelace area against another, and neither is read back from the routine that placed the tiles: the figures measure the tile polygon and the hole polygon that were actually drawn, and require them to add up to the cell. If a hole were drawn wrongly the sum would miss, and the generator refuses rather than printing a fraction nobody checked.

The same property from the other side #

The folded sheet reaches the same behaviour by a route with nothing in common.

Here there are no holes and no missing paper. The sheet is whole, every vertex carries its full turn, and the widening comes from the geometry of the zigzag: as the rows close, the columns pull in as well, so both in-plane dimensions fall together and both grow together on the way back out. That behaviour is inherited rather than designed — it belongs to the one vertex the pattern repeats, and the whole sheet has the single freedom that vertex has.

That last clause is the difference. The Miura’s two dimensions are two different functions of the fold state, and dividing one rate by the other gives an answer that depends on where in the motion the division was done.

Why one value moves and the other does not #

The two sheets differ in where the answer comes from, and it is worth being exact about it.

The Miura’s dimensions are obtained by solving a folding: the fold angles at each vertex are locked to one another, one state is chosen, and the sheet’s width and depth are then read off the solution. There is no reason for the two functions of the state to be related, and they are not. The value is a genuine output of a computation and it is whatever the computation gives.

The cut sheet’s dimensions come from a symmetry of the pattern, not from a solution. Rotate the whole cut pattern by a quarter turn and it maps onto itself, with the two directions exchanged. Whatever formula gives the width therefore also gives the height, with the same argument. So the two functions are not merely similar — they are the same function, and the ratio is minus one before anything has been calculated, before anything has moved, and, most of all, before the sheet was cut at all.

That is the surprising part. The value can be read off the flat sheet. It is a fact about a drawing on paper, established by a symmetry of the drawing, and the mechanism is only obliged to obey it.

Which theorem was checked, and how #

Three claims are asserted, and the design of each is meant to stop the figure agreeing with itself.

The hinges meet. Each tile’s corners are placed from that tile’s own centre, its own sense of rotation and its own angle, with no reference to any neighbour. Two tiles that share a hinge therefore arrive at that point along independent routes, and the check is how far apart they land. Over the whole array and the whole motion the worst separation is under one part in a hundred million million of a tile’s width — machine noise — which is evidence that the pitch is right rather than a restatement of it. Had the pitch been wrong by any amount at all, the corners would have missed and the sheet would have been one no scissors could produce.

The ratio is measured, not defined. Minus one is a suspicious number to find, precisely because it is the number the argument predicts. So it is obtained the same way the folded sheet’s is: two dimensions of the drawn array at two nearby openings, logarithms, a finite difference. The two measurements share no line of arithmetic with the claim being tested.

The contrast has to be real. The comparison is worth nothing unless the folded sheet’s value genuinely moves, so the figure requires that it move by more than a twentieth over the range drawn, and refuses to draw if it does not. A flat line beside a second flat line would be a picture of two measurements taken wrongly in the same way.

What the picture cannot show #

The hinge is the weakest part of all of this, and the drawing hides it completely.

In the model a hinge is a point: two tiles touching at a single location, free to rotate about it, transmitting whatever is needed and occupying no space. Nothing that can be made from a real sheet is like that. A blade has a width, and at the one place the pattern requires the sheet to be continuous, a cut of any width at all severs it. So a real cut sheet has hinges of finite size, made by stopping each slit short of the corner, and the ligament left behind is a small piece of sheet that has to bend. It bends, it fatigues, and it is invariably the first thing to fail. Everything above is geometry and the geometry has nothing to say about that; the vocabulary for it belongs to another subject, and this essay does not borrow it.

There is a second gap, and it is measurable — and it turns out to be exactly computable, which changes what it is a gap in.

A finite array with unequal sides does not return minus one. An eight-by-three array at twenty-five degrees comes out at −0.805, and a three-by-eight array of the same tiles at the same opening comes out at −1.243, which is its exact reciprocal.

The reason is that a finite array’s span is not a whole number of pitches. An $m$ by $n$ array measures $(m-1)P + a$ across and $(n-1)P + a$ down: one pitch for each gap between tile centres, and one tile’s width for the ends. Only the pitch depends on the opening, so differentiating and dividing gives

$$\nu = -\frac{n-1}{(n-1)P + a}\cdot\frac{(m-1)P + a}{m-1}.$$

At twenty-five degrees the pitch is $1.3289\,a$. Put $m = 8$, $n = 3$ into that and the answer is −0.8046; swap them and it is −1.2427. Both are the measured values to three figures, and the reciprocal relation between them is visible in the formula, since exchanging $m$ and $n$ inverts it.

And the deviation is about the array’s shape rather than its size. Set $m = n$ and the two factors cancel identically: a square array returns exactly minus one at every opening, whether it is forty tiles across or two. The forty-by-forty array quoted as the bulk case is exact not because it is large but because it is square, and a two-by-two array is exact for the same reason.

The other idealisations are the site’s usual ones. The tiles are rigid and never deform. The sheet has no thickness, so tiles that pass close to one another pass cleanly. And the tiles are perfect squares, which is why the pattern’s quarter-turn symmetry is available at all — the same construction with rectangles gives a value that is negative and is not minus one.

Which says what the finite-array number is measuring #

That formula is worth reading once more, because it says which of two things the eight-by-three figure is a fact about.

Rewrite it as a ratio of two corrections. Dividing top and bottom through gives minus one times $(1 + a/((m-1)P))$ over $(1 + a/((n-1)P))$ — one correction per direction, each of them the end tile’s width as a fraction of the span in that direction. The value is minus one exactly when the two corrections are equal, which happens when $m$ and $n$ are.

So the number is not measuring an edge effect in the usual sense, where the outermost row behaves differently from the interior. Every tile in this array behaves identically; there is no interior and no boundary layer. What the correction measures is that an array’s span contains one more tile-width than it does pitches, and the arithmetic of one more is felt more in the short direction than the long one.

That is worth separating because the two diagnoses suggest different remedies. An edge effect is fixed by making the array bigger. This is fixed by making it square, and a large non-square array does not converge to minus one: a hundred-by-three array at twenty-five degrees is still off by the same ratio of corrections, since the short direction’s correction stays large however long the other one gets.

It also says what to quote. The exact minus one belongs to the pattern, established by the quarter-turn symmetry, and it is the value a square patch measures at every size. Any other number a finite array returns is a fact about how the span was measured — where the ends of the array were taken to be — rather than about the mechanism. Both are honest measurements; only one of them is about the sheet.

What the holes cost #

Comparing the two sheets fairly means putting a price on each.

The cut sheet gains area and loses material. Fully open it covers twice the ground it started on and half of what it covers is nothing; a sheet needed to hold something out, or to keep something off, has lost the argument at that point. The folded sheet gains and loses nothing — all of its paper is still there, stacked rather than removed, which is why its shrink and its layer count are one quantity rather than two.

So the trade is legible. A fold buys the motion with thickness and a cut buys it with holes, and which is the better currency is entirely a question about the application. A stent wants holes. A solar array does not. What the two sheets share is the thing worth carrying away: neither property belongs to the paper, and both belong to the pattern drawn on it, which is why changing the pattern changes the behaviour without changing the material at all.

Who found it, and when #

The rotating-squares construction is Joseph Grima and Kenneth Evans’s, published in 2000 as a two-page note. Its claim is the one above: a plane of rigid squares hinged at their corners has an expansion ratio of exactly minus one, independent of how far it is open and independent of the size of the squares. The generalisations — rectangles, triangles, mixed tilings, three-dimensional versions — followed over the next decade, and most of them lose the exactness, which is a reasonable way of noticing that the exactness came from the symmetry rather than from the hinging.

The kirigami reading of it, in which the tiles are not separate objects at all but regions of one cut sheet, is later again and belongs to the 2010s, alongside the rest of the traffic between paper folding and hardware. It arrives from the manufacturing side: a laser or a die can put this pattern into a sheet in one pass, and the result is a component with a designed expansion that was never assembled from parts.

The lattice underneath the cut has a symmetry classification of its own, and that classification is crystal-symmetry.com’s subject rather than this one’s. Nothing here derives anything about it; the quarter-turn used above is a single observation about one drawing, made in order to identify two functions with each other, and it is the only symmetry argument in the essay.

Where the ladder goes next #

The rung below is the local statement: what a cut does to the angle at a point, and the fact that a slit which removes no paper does nothing to it at all. This sheet is the case where nothing is removed and everything changes anyway, which is why it belongs above that rung rather than beside it.

The rung above is the combination nobody has priced. A cut pattern and a crease pattern can be put in the same sheet, and the two mechanisms then share a motion — the tiles rotating while the creases between them close. What that costs, and whether the expansion ratio of the compound is anything as clean as minus one, is a real question and is not answered here.

Sideways, the interesting comparison is what a corrugation costs, where the currency is layers rather than holes, and what separates a panel from paper — because the cut sheet is the one structure on this site whose idealisation fails at a single identifiable place, and knowing which place that is turns out to be worth more than knowing that it fails.