The paper a pattern asks for

The Miura printed in this collection is not on a square. It is a sheet 170 mm across and 107 mm tall, and that proportion is not a layout decision — it is what the pattern asks for. Six columns, four rows, and a slant of twenty degrees fill a rectangle of about 1.59 to 1, and cutting anything else means either leaving paper uncreased or cutting the pattern in half.

The relation behind that number is short enough to write down, and it is one equation in four quantities: the columns, the rows, the slant, and the shape of the paper.

The equation #

A Miura cell is a parallelogram. Lay c of them across and the width is c cells plus the horizontal offset the slant introduces at the top — one cell height times the tangent of the slant. Lay r of them down and the height is r cells. On unit cells that is

proportion = (c + tan α) / r.

It is checked rather than asserted. The pattern is drawn from its vertex coordinates, its bounding box is measured, and the two agree to a part in 10¹⁶ on every size drawn — six sizes, from two by two to eight by six. The two computations share nothing: one lays out parallelograms and takes extremes of a coordinate list, the other is a tangent and a division.

The tangent term is the whole of what makes this interesting. Without it the proportion would be c/r, a ratio of integers, and any proportion with a rational value would be available. With it, the slant contributes an irrational shift to the width — so the proportion is a rational number only when the tangent is, and the equation stops being about counting.

Reading the equation #

Three terms, and each of them is a different kind of quantity, which is why the relation is worth stating rather than tabulating.

The columns are a count of cells, and they enter the width directly: adding a column adds a cell’s width and nothing else.

The rows are a count of cells too, and they enter the height directly for the same reason. So the ratio c/r is what a grid of rectangles would give, and it is the part anybody would guess.

The tangent is not a count. It is the horizontal distance the top of a column has moved relative to its bottom, in cell heights, and it is added once rather than once per column — because the zigzag reverses at every row, so the shear accumulates within a row and cancels between them. That single un-counted term is what makes the proportion irrational for almost every slant, and it is the whole reason a square is a special case rather than a matter of choosing counts.

Put the printed pattern through it: six columns, four rows, a slant of 0.35 radians, tangent 0.3654, giving (6 + 0.3654)/4 = 1.591. The sheet is printed 170 mm across, so it is 170/1.591 = 106.8 mm tall, which is the number on the printed sheet. The equation is not a description of the figure; it is what the figure was drawn from.

What a square costs #

Set the proportion to one and the equation becomes tan α = r − c.

The right-hand side is an integer, because the counts are. So a Miura fills a square exactly when its slant has a whole-number tangent, and the pattern is then one row taller than it is wide for a slant of 45°, two rows taller for 63.43°, three for 71.57°, and so on up an increasingly steep ladder of angles.

Those are real patterns and they fold. A three-by-four Miura at 45° is exactly square, as are four by five, five by six, six by seven and seven by eight — the whole family, since the condition depends on the difference of the counts and not on their size. A six-by-eight at 63.43° is exactly square too.

What they are not is the Miura anybody draws. The slant that makes a Miura behave like a Miura — that gives it a usefully negative Poisson’s ratio and a fold that closes in both directions at once — is well under forty-five degrees; this collection draws twenty. At twenty degrees the tangent is 0.3654, the six sizes drawn come out at 1.18, 1.12, 1.46, 1.59, 1.06 and 1.39, and not one of them is square.

The three ways out, and what each wastes #

A folder with a square sheet and a Miura in mind has three options and no fourth.

Cut a rectangle from the square. The pattern then occupies the whole of a smaller sheet, and the waste is what is trimmed off. At the sizes here that is between 15% and 37% of the paper: a six-by-four at twenty degrees uses 63% of a square’s width, throwing away better than a third.

Fill the square and take the counts that fall out. Choose the columns and rows so that the proportion is nearest to one — at twenty degrees, six by six is 1.06 and eight by eight is 1.05, both a few per cent off — and accept a strip of uncreased paper along one edge. This is the practical answer, and it is why square-sheet Miuras in the wild are nearly square rather than exactly so.

Change the slant. The proportion is continuous in α, so for any counts there is an angle that makes the sheet square; it is arctan(r − c), and for equal counts it is zero, which is not a fold. The angle is therefore chosen by the paper rather than by what the pattern is for — and the slant is what the Poisson’s ratio and the packing depend on, so this option buys a square by giving away the reason for folding a Miura.

Every one of those is a trade the pattern imposes rather than a preference. The equation has four quantities and three of them are decisions.

The slant is spent twice #

The reason the third option is expensive is that the slant is not free to begin with. It is the parameter that decides what a Miura does, and the paper’s proportion is asking for a share of it.

Take a four-by-five patch and sweep the slant. At 11.5° the proportion is 0.841 and the folded patch packs 12.6 layers deep, with a Poisson’s ratio of −0.12. At 20°, 0.873 and 10.2 layers at −0.33. At 45° — the square case — 1.000 and 7.95 layers at −1.40. At 63.4°, 1.200 and 9.01 layers at −2.65, with the crease length up by half.

So the square Miura at 45° is a real object with real properties, and they are not the shallow Miura’s properties. It packs a third less deeply, it costs a sixth more folding, and its in-plane response is four times stronger. Whether that is a better pattern depends entirely on what it is for: as a deployable that has to flatten, worse; as a material with a strongly negative ratio, better.

What the equation says is that these are not independent choices. A square sheet at a given size fixes the slant, and the slant fixes the mechanics. A designer who insists on square paper has, without noticing, chosen a Poisson’s ratio.

The other patterns choose too #

The Miura is not unusual in this. Every repeating pattern occupies a shape, and the shape is a consequence of the unit.

The Yoshimura is rows of diamonds, and the row height is not free — the big-little-big lemma caps the diamond’s half-angle at sixty degrees, so the cell is at most √3/2 as tall as it is wide and the sheet comes out a rectangle whose proportion is fixed by the counts alone. The waterbomb tessellation is built on a square grid and asks for a square when its counts are equal, which is the one pattern here that gets what it wants for free. The tapered corrugation is a rectangle by construction, and it is tapered precisely so that the two ends can differ; tapering across the folds instead puts Kawasaki’s two sums at 186.4° and 173.6° and the sheet will not close.

The twist tessellations are the exception that proves the rule, and only because they are cut rather than fitted: a patch of one is whatever the square happens to contain, and the units at the rim are cut through. A pattern that fits its paper and a pattern that is clipped to it are two different objects, and only the first has a proportion of its own.

What a square sheet admits #

Turned round, the equation answers the question a folder actually has: here is a square, what can go on it?

Exactly: any counts c and r with r > c, at the slant arctan(r − c). Nothing else fits without waste. The available slants are therefore 45°, 63.43°, 71.57°, 75.96° — an infinite ladder, but a rapidly steepening one, and every step of it is a different pattern with different mechanics.

Approximately, and more usefully: any counts at all, with a strip of uncreased paper left over. The strip’s width is the fractional part the equation leaves — for equal counts at a slant α it is tan α over c of the sheet, so it shrinks as the patch grows. At twenty degrees, four by four leaves 8.4% of the width uncreased, six by six leaves 5.7%, eight by eight leaves 4.4%.

That last sequence is the honest answer to why nobody notices the equation. On a large enough patch the leftover strip is a margin, and a margin looks like a margin rather than like a constraint. It is on the small patches — the two-by-two, the three-by-three, the sizes a page draws and a beginner folds — that the pattern’s demand for its own proportion is a third of the sheet.

The waste has a formula, and the slant is in it twice #

The leftover strip is quoted as a sequence of percentages, and writing it as an expression turns the third option from a resignation into a design rule.

Fill a square with equal counts $c = r$. The pattern’s box is $c + \tan\alpha$ wide and $c$ tall, so cutting a square of the larger dimension leaves a band of uncreased paper whose share of the sheet is

$$\frac{\tan\alpha}{c + \tan\alpha}$$

At twenty degrees that is 8.4 per cent at four by four, 5.7 at six, 4.4 at eight — the essay’s own three figures, exactly.

The slant appears in the numerator. So the waste falls not only with the patch size, which the essay notes, but with the shallowness of the slant, which it does not.

Which dissolves most of the conflict #

That matters because the shallow slant is what a Miura is for. Run the same expression at other angles.

At ten degrees the tangent is 0.176, so an eight-by-eight patch wastes 2.2 per cent of its square. At five degrees, 0.087, and the same patch wastes 1.1 per cent. A shallow, large Miura fills a square to within about one part in a hundred, and the strip left over is narrower than a margin anybody would cut anyway.

So the trade the equation imposes is severe only where it is asked for exactness. A Miura cannot fill a square exactly without a tangent of a whole number, which forces 45° and gives up the pattern’s mechanics; but it fills one approximately better and better as the slant shallows, and shallow is the direction a deployable wants for its packing and its Poisson’s ratio anyway.

The two demands therefore point the same way rather than against each other. The conflict the essay describes is real at the exact-square end and evaporates as soon as a designer will accept a per cent.

That also reads the three options differently. Cutting a rectangle wastes 15 to 37 per cent; changing the slant to 45° costs the mechanics; filling the square with equal counts at the slant the pattern wanted costs a few per cent of paper and nothing else. The third option is not the compromise — it is the answer, and the reason it looks like a compromise is that the equation’s exact solution is the one nobody should want.

Why the proportion is a design variable and not a nuisance #

Paper proportion has a literature of its own in this subject, and it is usually about the sheet rather than about the pattern. The A series keeps its shape when it is halved, which is a property of √2 and nothing to do with folding a tessellation; the square is a choice with consequences for how much flap the paper carries and where.

The Miura’s equation joins the two questions. Given a sheet, it says which patterns fit it; given a pattern, it says which sheet to cut. And it says that the two cannot both be chosen freely, which is the part that gets lost when a tessellation is drawn on whatever square is to hand.

There is a design reading of it as well. The difference r − c is what the slant’s tangent has to equal for a square, and a difference of one is the shallowest such pattern at 45°. Wanting a shallower slant on a square therefore means wanting r − c between nought and one, which no pair of integers gives — so the shallow square Miura does not exist at any size. The nearest approach is equal counts, whose proportion is 1 + tan α over c, tending to a square as the patch grows: 1.09 at four by four, 1.06 at six by six, 1.05 at eight by eight. The pattern gets squarer by getting bigger, which is a boundary effect wearing a different hat.

The sheet has been the variable before #

This is the third time in this collection that the paper’s shape has turned out to be part of the mathematics rather than part of the presentation, and the three are worth putting together because they come from different directions.

The A series keeps its proportion when it is halved, which makes √2 the one rectangle that folds into copies of itself — a property of the sheet alone, with no pattern involved. The silver and bronze rectangles are the rest of that family, each self-similar under a different number of folds. Those are answers to what shape of paper is special, asked without reference to what is going to be folded on it.

Which reference points a sheet admits depends on its proportion, which is the middle case: the pattern is not fixed, but the sheet decides what can be constructed on it, and a square and an A4 sheet reach different sets of points in the same number of folds.

The Miura’s equation is the third and the most specific: a named pattern, at a named size, asking for one proportion and no other. Read the three in order and the sheet moves from being the object of study, to being a constraint on construction, to being an output of the design — and the last is the one a folder meets, because it is the one that says which paper to cut before anything is folded at all.

Where the equation stops #

It is exact for the pattern and approximate for the paper. The bounding box of the crease pattern is what the equation gives. A folder cuts a sheet with a margin, and the margin is not in it.

It assumes the cell is a unit square before shearing. A Miura’s cell has two lengths and the ratio between them is free in the way the sector lengths are always free — a cell twice as wide changes the width term and not the height. The equation generalises by carrying the cell’s own proportion as a factor; drawn here with equal cell sides so the two counts read directly.

It says nothing about which of the two counts is the width. A Miura and the same Miura turned through a right angle are the same pattern on sheets of reciprocal proportion, so every row of the table has a partner. Which way round it is drawn is a genuine choice, and it is the one thing here that really is presentation.

And it is about a pattern that fills its sheet. Nothing forces a Miura to reach the paper’s edges. A patch drawn small in the middle of a large square has whatever proportion the drawing gives it, which is a picture of a Miura rather than a Miura on that sheet — and the distinction is the same one between a tessellation and a patch of one.