A sheet is as large as two arms

A sheet has a size as well prices the second of the substrate’s constraints. A model’s folded footprint times its mean layer count is the area of the paper, so the linear shrink is the square root of the layer count, and a sixty-four-layer model finished at a hand’s width wants more than a metre of sheet.

That essay treats the sheet size as a demand and says nothing about the supply. The supply has a bound too, and it is not a bound on thickness.

A hand-made sheet is formed by dipping a mould — a frame with a screen stretched across it — into a vat of pulp, lifting it out with a layer of slurry on it, and shaking it so the fibres interlock. What comes out of the vat is a wooden frame plus a sheet of paper that is almost entirely water, and somebody has to hold it and move it, several hundred times a day. The largest sheet a tradition can make is bounded by what a person can lift and shake, which is a bound in area and has nothing to do with how thin the paper is.

The arithmetic, and what it rests on #

The mass on the mould is the frame plus the sheet plus the water in it.

The frame is the mould’s own weight, which scales with its area — a bigger mould needs a stouter frame and more screen, so it is reasonable to write it as a weight per square metre. A hand mould is a wooden frame with a bamboo or brass screen, and anywhere from four to twenty kilograms a square metre covers what such a thing weighs.

The sheet is its grammage — forty grams a square metre for a good washi — and the water is several times its own mass, because a newly couched sheet is mostly water. Ten times is the figure used here.

So the load per square metre is the frame’s weight plus about half a kilogram, and the largest sheet is the area at which that reaches what a papermaker can carry.

Every number in that table is uncertain by a factor of two and the range as a whole is not. The frame’s weight is not known to better than that; the sustainable lift is a judgement rather than a measurement; and the water ratio depends on the pulp and the draining. What survives all of it is the bracket: the largest hand-made square sheet is somewhere between half a metre and two metres a side.

Which is exactly the demand #

The reason to compute a bracket that loose is what it brackets.

A model finished at a hand’s width — a hundred and fifty millimetres — needs a sheet of $150\sqrt{L}$ for a mean layer count of $L$. Four layers wants three hundred millimetres. Sixteen wants six hundred. Sixty-four wants twelve hundred. A hundred and twenty-eight wants seventeen hundred.

Three of those four land inside the bracket, and the fourth is above its top. So the question can a sheet that large be made? is not a question with an obvious yes, and it is not a question with an obvious no either. The supply and the demand are the same sizes, which is the only claim this arithmetic is strong enough to make and is a claim the substrate sequence has not made before.

That matters because the essays before this one treat the sheet as available. The paper had to arrive first argues that the complex tradition is downstream of thin paper, and its evidence is thickness throughout. Which ceiling is binding computes a crossing between the stack and the hand’s precision, and both of its bounds improve with a larger sheet — so its conclusion, that the substrate is the limit on classical papers, is drawn on the assumption that the sheet can be whatever size the argument wants.

It cannot. The sheet is made by a person, and the person’s reach is the ceiling on that assumption.

A bound in area is a bound that squares #

The bound is on area and the demand is stated as a side, and the difference between those two is what makes the constraint bite harder than it looks.

Doubling a model’s finished size quadruples the paper it needs at a fixed layer count, and a mould that carries four times the area has to carry four times the load. So a tradition that can make a sheet six hundred millimetres square and wants one twelve hundred millimetres square needs four times the lift, not twice — which the table shows as the gap between the 5 kg and 20 kg columns being a factor of two in side and four in area.

The same squaring runs the other way and is what makes small models attractive. Halving the finished size quarters the paper, so a sixty-four-layer model at seventy-five millimetres wants a sheet of six hundred rather than twelve hundred, and lands comfortably inside every column of the table.

So the two variables a designer controls — the layer count and the finished size — enter the sheet’s side at the same power. $150\sqrt{64}$ and $300\sqrt{16}$ are the same sheet. A model with four times the layers at half the size costs exactly what the original did, which is a substitution nobody in these essays has had a reason to state and which follows from the shrink argument immediately.

Why the bound is in area and not in thickness #

This is the part worth carrying past the numbers.

Every other constraint in these essays improves as the paper thins. A thinner sheet makes a thinner stack, so the layer ceiling rises. A thinner sheet also weighs less, so — one might think — a larger sheet could be lifted.

It could, a little. But the sheet’s own mass is half a kilogram a square metre against a frame of four to twenty, so the frame dominates the load by a factor of ten to forty, and halving the paper’s weight changes the liftable area by a per cent or two. The water is worse: at ten times the sheet’s mass it is four and a half kilograms a square metre for an eighty-gram sheet and two for a forty-gram one, which is still small beside the frame.

So the size bound is essentially independent of the paper being made on it. A tradition that improves its paper from a hundred microns to forty has changed nothing about how large a sheet it can produce, and the two constraints on a design — layers and size — have moved in the ratio one-to-nothing.

That breaks these essays’ own story. The story is that thinning the paper raises the ceiling; what the thinning raises is one ceiling of three, and the one it does not raise is the one a design of many layers runs into first once the finished object has to be larger than a thumbnail.

What a tradition would notice #

A constraint that binds shows up in what a tradition makes, and this one predicts something specific.

If the limit were thickness alone, a tradition improving its paper would produce models of steadily more layers at a steady size, because nothing else would have changed. If the limit is the sheet, improving the paper produces models of more layers at a smaller size — because the layer count and the finished size trade at the same power, and only one of the two has got cheaper.

So the prediction is that a tradition with good paper and ordinary moulds makes small complex things, and a tradition with large sheets makes large simple ones. Neither is a statement about skill, which is what makes it worth writing down: a reader looking at a small intricate model and a large plain one will read a difference in ambition, and the arithmetic says it may be a difference in what came out of the vat.

That is a claim the arithmetic here cannot check, for the reason the first of these essays gives about its own field: the evidence would be surviving models with their sheet sizes recorded, and sheet sizes are the one thing a finished model does not carry. What it can do is say which measurement would settle it, and it is not a measurement of paper thickness.

The shape of the escape #

There is an escape and the tradition took it, which is worth naming because it is the evidence the arithmetic is describing something real.

Sheets can be joined. A larger sheet than a mould makes can be assembled from smaller ones, and the join is a line of glued overlap — thicker than the paper, visible, and a place the crease pattern has to be planned around. That is a real technique and it has a cost: the sheet has a thickness is the constraint a join doubles locally.

The mould can be worked by two people. The twenty-kilogram column of the table is two pairs of arms, and the largest traditional sheets are made that way. It buys a factor of two in load, which is a factor of $\sqrt{2}$ in linear size: a metre becomes one and a half.

Or the model can be smaller. A sixty-four-layer model at a hand’s width wants twelve hundred millimetres; at half a hand’s width it wants six hundred. The shrink argument runs both ways, and the tradition of very small complex models is the arithmetic taken in that direction.

Each of those is a way of paying the size bill rather than of removing it, which is what a real constraint looks like from the inside.

What the arithmetic does not establish #

No date, and no place. Nothing here says how large a sheet any tradition actually made. What it says is what the arithmetic of a mould implies about the range, and the range is wide. A historical claim would need surviving sheets and surviving moulds, which is the kind of evidence this field’s other arguments rest on and which this one deliberately does not use.

The frame’s weight is a guess with a range round it. The whole result is stated as a bracket for that reason, and every conclusion drawn is a conclusion about the bracket rather than about a number in it.

Shaking is not lifting and is not modelled. The sheet is formed by agitating the mould as the water drains, which is a dynamic load several times the static one and is what actually limits a papermaker’s day. Treating the constraint as a static lift understates it, so the bracket is if anything generous.

And a mould is not a square. Moulds are made in the proportions the paper is sold in, which are not square, so “the largest square sheet” is a convenience. A rectangle of the same area has a longer side and a shorter one, and what a design needs is usually the shorter one.

And nothing here is a material property. Four things that are not true lists the idealisations the theorems of this subject rest on, and none of them is about how the paper was made. The lift bound is not a fifth idealisation; it is a fact about the supply of sheets, which sits outside every theorem and decides which of them a folder is ever in a position to use.

Nothing here is about machine-made paper. A paper machine makes a continuous web and its width is bounded by engineering rather than by arms; the whole argument applies to the hand-made era and stops at the point the era does.

The bound is on the supply, not on the design #

One distinction is worth drawing before the limits, because it separates this essay from the two before it.

The stack ceiling and the grid ceiling are bounds a design meets: a folder with a given paper and a given sheet runs into them by making the design more elaborate. This one is a bound on what can be obtained. A folder does not meet it by folding; they meet it at the shop, and no amount of skill or ambition moves it.

That makes it the only constraint in these essays that is outside the folder entirely, and it is why it has a different kind of evidence behind it — arithmetic about a mould rather than about a model.

How the figures were drawn #

The lift curve is one division. Load per square metre is the frame’s weight plus the sheet’s grammage times one plus the water ratio; the largest area is the lift divided by that; the side is its square root. There is nothing else in it.

The bracket is checked rather than described. The figure stops if the ratio between the largest and smallest sheet across the whole range of frames and lifts is under two or over eight — the first would mean the constants do not matter and the range is a number, the second that they matter so much the range says nothing.

And the demand lines are required to land inside it. If the sheet a model of that size needs at every layer count drawn fell outside the bracket, the figure would have nothing to compare and would stop.

Still open: the proportion of the mould #

The arithmetic bounds an area and a design needs a side, and the two differ by the proportion.

A mould in a two-to-one proportion of the same area has a long side a factor of $\sqrt{2}$ longer than a square one and a short side a factor of $\sqrt{2}$ shorter. A design needing a square sheet gets the short side; a design that can use a long strip gets the long one. So the shape of the sheets a tradition makes is a design constraint on top of the size, and it is the one the proportion sequence is about from the geometry’s side and nobody has connected to the vat.

The measurement that would settle it is a survey of surviving moulds rather than a calculation, and what a calculation can contribute is the trade: for a fixed lift, every proportion of the same area is available, so the proportion a tradition settled on was a choice and not a bound. Which means the proportions paper is sold in are evidence about what it was for, and the argument would run from the mould to the designs rather than the other way.

And the joins deserve their own arithmetic. A sheet assembled from four smaller ones has three seams, each of double thickness, and the layer count a design reaches is limited by the thickest point of its stack — so a join running through a region of many layers costs more than one running through a flat region. Where the seams should go for a given crease pattern is a placement problem with a stated objective and nobody has posed it.

Sideways from here, the size bound explains a practice the thickness argument does not. A model made from a joined sheet, or from a sheet at the top of the bracket, is an expensive object before it is folded — so the very large complex models are objects made once, by somebody who could obtain the paper, which is the same economics the two traditions had to negotiate when they met. A tradition’s characteristic scale is partly a fact about its vats.

The habit worth carrying is about constraints that do not share a variable. When two bounds on a design improve with different properties of the same material, check whether the one being improved is the one that binds. Thinning the paper is the whole of this subject’s story about its own materials, and it does nothing at all to the size of the sheet — which is the bound a model of many layers meets first as soon as it has to be large enough to look at.