Eighty layers and the sheet decides the rest

The five essays before this one found three bounds between them, and none of the three bounds a design on its own.

The stack says a layer count times a thickness has to stay under the smallest feature a folder works to, which bounds layers given a paper and says nothing about which paper. The sheet says the finished footprint times the mean layer count is the paper’s area, which bounds a finished size given a sheet and says nothing about which sheet. The crease floor says a paper thinner than a fibre and a half will not hold a fold, which bounds the paper and says nothing about anything else.

Put in that order they close, and what they close on is a region.

Three numbers, and the region they leave #

The floor fixes the paper. A fibre is about twenty-five microns across and a crease needs a fibre and a half of paper through the thickness to have anything to hinge, so the thinnest usable sheet is about thirty-eight microns. That is washi, and it is the thinnest paper in general use for complex folding, which is the whole of the evidence for the number.

The paper fixes the ceiling. Thirty-eight microns against a three-millimetre stack is eighty layers, and it is the largest layer count any unbacked paper allows at any size whatever. A thicker paper allows fewer; a thinner one is not a paper a crease holds in.

And the sheet falls away as the square. A model finished at $f$ millimetres and $L$ layers needs a sheet of $f\sqrt{L}$, so a sheet of $S$ allows $L \le (S/f)^2$. At the middle of the lifting bracket — twelve hundred millimetres — that is eighty layers at a hundred and thirty-four, thirty-six at two hundred, sixteen at three hundred, four at six hundred.

The two change places at a hundred and thirty-four millimetres, which is a hand’s width, and the crossing is $S/\sqrt{L_{\max}}$ — three numbers from three of these essays and nothing else.

What is on each side of the corner #

The corner divides the field into two regimes that want completely different things, and neither of them is the regime these essays’ first essay is about.

Below the corner the paper binds, and a model there is limited by how thin the sheet is. Better paper buys more layers directly; a larger sheet buys nothing at all, because the design already fits. That is the regime a small intricate model lives in, and it is where the first of these essays’ argument — that the complex tradition is downstream of a manufacturing achievement — is exactly right.

Above the corner the sheet binds, and a model there is limited by how much paper there is. Thinner paper buys nothing whatever: the stack is not what stops it. A larger sheet buys layers as the square, so going from a six-hundred-millimetre sheet to a twelve-hundred quadruples the layer count at any finished size.

That second sentence is the one the earlier essays here have not said. For a model larger than a hand’s width, improving the paper is a waste of effort, and every argument in this field about thin paper is an argument about the smaller half of the map.

Halving the sheet moves the corner to sixty-seven millimetres and quarters the allowance everywhere above it. A tradition whose vats make half-size sheets is not half as capable; above the corner it is a quarter as capable, and below the corner it is exactly as capable, because the paper is doing the work there.

The third of these essays’ crossing, and this one #

Two crossings now exist in these essays and they are not the same crossing, which is worth separating because they are both about which bound binds.

Which ceiling is binding crosses the stack against the hand — how fine a division a folder can place — and finds the crossing moving with the paper’s thickness: on classical papers the stack binds at every size worth using, and only at tissue weights does the hand take over. The crease floor then shows that the tissue weights are below the floor, so that crossing is almost entirely outside the region a folder can work in.

This one crosses the stack against the sheet, and it is inside the region everywhere. The difference is which quantity the second bound is a function of: the hand’s precision does not depend on the finished size at all, so its crossing with the stack moves only with the paper. The sheet bound falls as the square of the finished size, so it crosses the stack somewhere whatever the paper is.

A bound that varies with the quantity being plotted always crosses; a bound that does not, crosses only if the constants happen to put it there. That is why the third of these essays’ crossing turned out to be nearly empty and this one is not, and it is a distinction worth making before computing any crossing at all.

Why eighty is a ceiling and not a target #

The number to carry is the eighty, and it is worth being precise about what kind of statement it is.

It is not a claim that no model has more than eighty layers. Models have regions of many layers and regions of few, and the stack argument is about the thickest point — the first of these essays computes sixty-four layers at the thickest point of a model as six and a half millimetres in copier paper, and a model’s mean layer count is far below its maximum.

It is a claim about the thickest point. Eighty layers of a thirty-eight-micron paper is three millimetres, which is where a fold stops being a fold and becomes a bend in a block. Past that the paper does not crease, the crease does not stay, and the layers slide.

And it is a ceiling nothing on this map moves. A larger sheet does not move it. A thinner paper is not available. A better folder does not move it either, which is the part that makes it worth stating: every other constraint in this collection is a constraint on a design or a construction, and this is a constraint on the object.

Where the deployables sit #

The same three bounds apply to anything folded from a sheet, and a structure that is not paper sits somewhere else on the map — which is a useful check that the map is about materials rather than about paper.

A deployable membrane is a polymer film a few microns thick with no fibres in it, so the crease floor does not apply and the stack ceiling is enormous: at five microns a three-millimetre stack is six hundred layers. Its sheet is made by a machine rather than by arms, so the sheet bound is somewhere else entirely. Both of the bounds that close this region are bounds on hand manufacture, and a structure made by other means is not in the region at all.

What replaces them is a different pair. A deployable’s crease count is a reliability budget — every hinge has to work and the probability is a product — and the fold count sets the spring, because a hinge that has to open again cannot be a crease. So the same design variable, the layer count, is bounded from above by two entirely different arguments once the object stops being paper and stops being folded once.

That is worth stating because it says what kind of result this map is. It is not a fact about folding; it is a fact about folding a hand-made sheet by hand, and the ceiling of eighty is the ceiling of a craft rather than of a geometry.

The region has a shape nobody designs for #

Read as a design space, the region says something about what is worth attempting, and the shape is not symmetric.

Along the bottom — few layers, any size — everything is available, and nothing in these essays constrains a simple model at any size the vat can supply.

Up the left — many layers, small size — the region is a flat ceiling at eighty, so a folder working small can add layers freely until the stack stops them, and the finished size does not enter.

And the corner is where both bind at once, which is the only point on the map where a design is using both of its resources fully. A model at a hand’s width with eighty layers is the most that can be made of a hand-made sheet, in the precise sense that every other point of the region wastes one bound or the other.

That is a statement about a tradition’s characteristic object, and it is close to being a description of one. A complex model in the classical tradition is a few hands across with a few dozen layers at its thickest, which is a point near the corner. Nobody arrived there by computing this, and the arithmetic says it is where the constraints meet.

Reading a tradition’s position off its vats #

The map turns two manufacturing facts into a prediction about what a tradition can make, and the prediction is sharper than either fact alone.

Give a tradition a thinnest usable paper $t$ and a largest sheet $S$. Its ceiling is $3000/t$ layers, its corner is at $S\sqrt{t/3000}$ millimetres, and everything it can fold is under those two lines. Two numbers from the mills fix the whole design space.

So two traditions with the same paper and different vats do not differ in what they can fold small — they differ only above the corner, and the difference there is a factor of the sheet sizes squared. Two traditions with the same vats and different papers differ everywhere below the corner and nowhere above it. Where two traditions met is a question about notation and practice, and this says what would have been visible in their objects: the merge of two paper supplies is a merge of two regions, and the union is larger than either in one direction only.

That is a testable claim about surviving models, and it needs their sheet sizes, which is the datum nobody recorded. What the map contributes is knowing which datum to want.

What the region leaves out #

It is three numbers with brackets round each of them. The floor is a reading off six papers; the stack’s three millimetres is a folder’s working figure rather than a constant; the sheet is the middle of a bracket that runs from about five hundred to two thousand millimetres. Every line on the map moves if any of them does, and what does not move is that the two bounds cross and that the crossing is at $S/\sqrt{L_{\max}}$.

Mean layer count and maximum layer count are used as though they were one number. The sheet bound is about the mean — the folded footprint times the mean is the area — and the stack bound is about the maximum. A design whose thickest point is far above its mean sits worse against the stack and better against the sheet than the map says, and every real design is such a design.

No joins. A sheet can be assembled from smaller ones, which moves the sheet bound at a cost in local thickness, and nothing here prices that.

And no laminates. A foil-backed sheet evades the crease floor entirely and therefore has no ceiling of this kind; the whole map is about unbacked paper, which is the material the tradition the essays here is about actually used.

The one number the region does not contain #

Everything on the map is a bound on a design and none of it is a bound on a folder, which is the omission worth naming before the limits.

The hand appears once in these essays, as the finest division a pair of hands can place, and the third of these essays’ crossing is the one place it binds. The crease floor takes almost all of that away. So the map that comes out has no term in it for skill at all — a beginner and a master face the same eighty layers and the same corner, and nothing about either line moves with practice.

That is not a claim that skill is irrelevant. It is a claim about what kind of quantity these three bounds are: they are properties of the paper and of the vat, and a folder meets them as facts rather than as difficulties. What skill decides is how close to the region’s edge a design can actually be taken, which is a different question and is not one a substrate argument can reach.

What is computed #

Three divisions and one square root, which is worth saying because a region this consequential ought to look harder to compute than it is. The floor is the fibre width times one and a half; the stack ceiling is the feature size divided by the floor; the sheet ceiling is the sheet divided by the finished size, squared; and the corner is the sheet divided by the square root of the stack ceiling.

The crossing is checked rather than described. The figure stops if both bounds do not bind somewhere on the range drawn — a map on which one of them never binds would be a map with one bound on it and would not be this argument.

And the ceiling is checked too. No finished size may reach more layers than the stack allows, which is a check that the minimum is being taken and not the maximum — a figure drawing the maximum would show a region opening upward as the model gets smaller, which is what the sheet bound alone says and is exactly the reading the other two of these essays exist to prevent.

Still open: the mean against the maximum #

The map treats a design as a single layer count and a design is a distribution.

The two bounds read different statistics of the same design. A sheet is consumed by the mean and a fold is stopped by the maximum, so the honest object is a pair of numbers with a ratio between them, and the ratio is a property of the crease pattern. A uniaxial base with a thin, deeply layered spine and broad flat flaps has a high ratio; a tessellation has a ratio near one. Computing that ratio for the printed patterns is a measurement that could be made from what is already here — the layer census is run on several of them — and it would turn the single map into a family of maps, one per kind of design, with the tessellations near the corner and the bases far from it.

And the corner suggests a design criterion nobody uses. A design sitting exactly at the corner uses its sheet and its paper equally, and any design away from the corner is wasting one of them. So a designer given a sheet and a paper has a preferred finished size — $S / \sqrt{L_{\max}}$ — at which nothing is left over, and it is computable before anything is drawn. Whether that is a useful thing to know or merely a true one depends on the ratio above, which is why the two questions are one question.

The habit worth carrying is about bounds that each leave a free variable. A set of constraints that individually bound nothing may collectively bound everything, and the way to find out is to eliminate the free variables against one another. Five of these essays each ended with a bound and a shrug; the shrugs were the same variable appearing in two of them, and substituting closes the region in one line.