A sheet has a size as well

The paper had to arrive first makes an argument that does not depend on a document, which is unusual in this field. A model with sixty-four layers at its thickest point, folded in ordinary copier paper, is six and a half millimetres of stack. Stack thickness is layer count times sheet thickness, a fold stops working when the stack approaches the smallest feature being folded, and so the layer count a design can reach is fixed by the substrate rather than by the folder.

That is one constraint and it is the one the thinness of the paper decides. There is a second, it comes from the same conservation law this collection runs on, and it is decided by the paper’s size.

The shrink is a square root #

The paper is all still there: the folded footprint times the mean number of layers over it is the area of the sheet, exactly. It is a conservation law, it has no slack in it, and its consequence for size is immediate.

Write L for the mean layer count and S for the side of a square sheet. The folded footprint has area S²⁄L, so its side is S⁄√L. The linear shrink is the square root of the layer count.

So a design that piles four layers finishes at half the sheet’s width, one that piles sixteen at a quarter of it, and one that piles sixty-four at an eighth. Turned round: a finished size of f at L layers needs a sheet of side f·√L.

At a hand’s width — call it a hundred and fifty millimetres — that is three hundred millimetres for four layers, six hundred for sixteen, twelve hundred for sixty-four and seventeen hundred for a hundred and twenty-eight.

Which sheets those are #

An A0 sheet is 841 by 1189 millimetres. Every A-series size is a halving of the one above, so the longest dimension available in the series is A0’s 1189.

Put the requirements beside them. Four layers wants 300 and A4’s 297 is a whisker short, so A3 or A2 will do. Sixteen wants 600, which is A1. Thirty-two wants 849, which is A0. Sixty-four wants 1200, which is past A0 by eleven millimetres, and a hundred and twenty-eight wants 1697, which is past it by half as much again.

That is the finding stated at one finished size, and the finished size is a choice — a smaller model needs less paper, in exact proportion. What does not change with the choice is the ratio: the sheet is always √L times the model, so a design’s layer count fixes how much larger the paper has to be than the thing that comes out of it.

Which turns the substrate argument into a two-parameter one. A tradition of complex folding needs paper that is thin enough and large enough, and the two are different manufacturing achievements. Thinness is about the beating and the fibre; size is about the mould and the vat, and a mill that can make one need not be able to make the other.

What the mean layer count is, and why it is not the maximum #

The arithmetic uses the mean layer count and the thickness argument uses the maximum, and running the two together would be a mistake worth naming.

The stack bound is about the thickest point. A model with sixty-four layers somewhere and two layers over most of its area has a six-millimetre lump in one place, and it is that lump that makes the fold stop working — the place a model is twice as thick where it is thickest.

The size bound is about the average. The same model’s footprint is decided by the total paper divided by the mean, and if most of it is two layers deep then the mean is near two and the model finishes at nearly the sheet’s own width.

So a design can be size-cheap and stack-expensive, or the reverse, and the two bounds are genuinely independent. A uniformly folded design has its mean equal to its maximum and pays both bounds at once; a design with one deep region pays the stack bound at that region and the size bound at its average.

That gives a testable shape to the historical claim. If the constraint that mattered were the stack, designs would avoid deep local piles; if it were the size, they would avoid high averages. Those are different design pressures and they would leave different marks on a repertoire — and separating them in the record is beyond what this collection can do, though it is the right question.

The ratio nobody quotes #

Every published model has a finished size and most have a recommended sheet size, and the quotient of the two is a number the field carries everywhere and never states.

It is the square root of the mean layer count, which makes it a measurement of the design rather than of the paper. A model recommended at a twenty-centimetre sheet and finishing at ten centimetres has a mean layer count of four. One finishing at five centimetres from the same sheet has a mean of sixteen.

That is a measurement anybody can make from a published diagram’s first page, and it does not require the model to be folded. It is also more robust than the maximum layer count, which requires counting layers at the worst point of a finished model — an awkward measurement that damages the specimen.

So the size ratio is a cheap proxy for a quantity the field cares about and does not measure. Whether the proxy is any good depends on how far a real model’s footprint departs from the conservation identity, and it departs in one direction only: a folded model has an outline that is mostly crease where paper does not lie flat, so its footprint is larger than the identity says and the inferred layer count is an underestimate.

An underestimate in a known direction is a usable measurement, and this is one.

What the record can and cannot be asked #

The arithmetic gives a requirement and the field’s rule is that a historical claim needs a source, so it is worth being exact about which half of this is which.

Computed here: that a design of a stated layer count finished at a stated size requires a sheet of a computed size. That is a consequence of a conservation law and needs no document.

Not computed here: what size sheets any mill in any tradition was making at any date. Every one of those is a documentary question, this collection’s record carries dates for when paper was made in three places and nothing whatever about sheet dimensions, and inventing a number would be exactly the failure this field was built to avoid.

What the arithmetic does supply is a question the record could be asked, which is more useful than an answer it cannot give. Surviving sheets have dimensions; moulds survive; paper sizes are recorded in trade documents for the periods where trade documents survive. A historian with those could put a curve of available sheet size against date beside the curve computed here, and the crossing would be a claim with evidence behind it.

That is the shape this field’s arguments have to take, and the first rung of this ladder is the model. It priced a stack in millimetres and left the question of when paper of that thinness existed to the sources; this rung prices a sheet in millimetres and does the same.

The value of a computation in a field like this is that it turns a vague claim into a measurable one. “Complex folding needed better paper” is not checkable. “A sixty-four-layer model at a hand’s width needs a sheet 1,200 millimetres across” is checkable the moment somebody has a list of sheet sizes, and the arithmetic is exact.

Why the square root is the interesting exponent #

A square root is a weak growth and it is worth noticing that this is the good case rather than the expected one.

Suppose the shrink went as the layer count rather than as its square root — that a sixty-four-layer model finished at a sixty-fourth of the sheet’s width rather than an eighth. Then a hand-sized model at sixty-four layers would need nine and a half metres of paper and the tradition would be impossible.

It does not, and the reason is that folding shrinks in two directions. The layer count is an area ratio and the size ratio is a length ratio, so the exponent is a half — and that halving is what makes complex folding possible at all with sheets a mill can make.

Which puts the substrate argument in a slightly different light. The first rung says the tradition is downstream of thin paper; this rung says it is downstream of large paper too, and adds that the paper requirement grows slowly enough for the demand to have been met. A design’s ambition costs the square root of itself in sheet, and a tradition that had to pay linearly would have stopped at four layers.

Reading a repertoire by its sheets #

If a design’s ambition costs the square root of itself in paper, then a tradition’s recommended sheet sizes carry information about its designs, and that is a record that survives rather better than the designs do.

The instruction is nearly always on the first page: fold from a square of a stated side. That number, divided into the finished size when the finished size is given, is the square root of the mean layer count — so a shelf of published models is a distribution of mean layer counts that nobody has plotted.

What such a distribution would show is how the ambition of a repertoire moved, in a quantity that is a property of the designs rather than of what anybody said about them. A tradition whose recommended sheets got larger relative to its finished models was folding deeper, whatever it called itself.

And it is a measurement with an unusual property for this field: it does not need the model to survive. A diagram sheet with a size instruction and a finished photograph is enough, and diagram sheets survive far better than folded paper does — which matters, because what this field’s record mostly lacks is objects.

That is the measurement this rung is really pointing at and cannot make. It needs a shelf and a ruler rather than a computation, the arithmetic that licenses it is the identity above, and the answer would be a curve of design ambition against date computed from a source that exists.

Which theorem was checked and how #

The identity is checked at every row rather than cited. The sheet each layer count needs is computed as the finished size times the square root, and the areas are then required to balance — a footprint of the finished side carrying that many layers must hold exactly the area of a sheet of the computed side, to within a part in a trillion.

Some layer count must want a sheet larger than any offered. The figure refuses a comparison in which every requirement fits, because the essay’s point is that the bound is on the page rather than off it.

The growth is asserted across the range. The largest requirement must exceed the smallest by more than a factor of three, so a comparison over a narrow band of layer counts — where the bound would look like a detail — cannot be drawn.

And no sheet size here is a historical claim. The A-series dimensions are a modern standard with a definition; what the essay says about mills is attributed in the prose and is not computed anywhere.

Where the model stops #

The sheet is square and paper is not. An A-series sheet is 1 by √2, so a design needing a 1200-millimetre square has to be cut from something larger still. That makes the bound tighter rather than looser.

The footprint is taken as the identity gives it. A real folded model has thickness, so its layers do not lie on one another exactly and its footprint is larger; and it has open regions where no paper lies at all, which enlarges it further. Both push the same way and both mean a real model is bigger than this arithmetic says.

The mean layer count is a property of the finished model and is rarely published. What is published is the maximum, when anything is, so applying this arithmetic to a real design usually means inferring the mean from the size ratio rather than the other way round.

And nothing here is a claim about what any mill could make. The essay computes what a design would require; whether a particular tradition had access to a sheet of that size at a given date is a documentary question, and this field’s rule is that a documentary question is answered by a source or not at all.

What the picture cannot show #

The bars show what a design needs and cannot show what any design was. No model is measured here, no layer count is taken from a specimen, and the six counts drawn are a range chosen to span what complex designs plausibly reach.

Nor can the figure show the interaction with the stack bound, which is the other half of the substrate’s constraint. A design at the top of this chart may be well inside the thickness bound or well past it, and the two are drawn separately because they are decided by different properties of the same model.

The clearest absence is the shape. A model that finishes at a hand’s width is not a square of that side; it is a figure with limbs, and its bounding box is much larger than its folded footprint. So the sheet a design really needs is larger than the arithmetic here by whatever the shape’s inefficiency comes to, which is the packing question the design field spends a ladder on.

The idealisation, named #

The sheet has zero thickness and the folded state is flat, which is what makes the footprint times the mean layer count exactly the area.

Give the sheet a thickness and the identity acquires a correction: the layers stand off one another, the pile has a depth as well as a count, and the footprint grows. The correction is small for a few layers and is not small for sixty-four, which is precisely the regime this ladder is about.

So the two bounds interfere at the top of the range. A design deep enough to be interesting for the size bound is deep enough that its own thickness is enlarging its footprint, and the sheet it needs is larger than √L times the finished size by an amount that grows with L.

Both idealisations point the same way, which is the useful thing about them. Every physical correction makes the paper requirement larger, so the numbers here are a floor on what a design needs rather than an estimate of it.

Where the ladder goes next #

Two bounds have now been priced and both are ceilings on the same design, so the next question is which of them binds — and the answer turns out to depend on the sheet in a way that inverts the ladder’s own story.

The stack ceiling improves as the paper gets thinner. The grid ceiling improves as the sheet gets larger. They cross at a sheet size that rises as the paper thins, so on the papers a classical folder had, the substrate really is the limit at every size worth using — and only at tissue weights, on small sheets, does the limit stop being the paper and become the hand. Which ceiling is binding computes the crossing and finds it landing, for the thinnest papers, at about the size of a sheet somebody would actually use.

The habit worth carrying is about arguments from a material. A substrate usually constrains a design in more than one way, and the ways are decided by different properties of it. Thickness and size are both properties of paper, they bound different things about a model, and an argument that names only one of them has found a bound rather than the bound.