Which ceiling is binding

A sheet has a size as well adds a second constraint to this ladder’s first, and both of them are about the same design. The stack limits the layers; the sheet limits the size; and neither says which of them a real design is up against.

That question has an answer, it is arithmetic, and the answer runs the opposite way from the intuition.

Two ceilings on one number #

Both constraints bound the layer count L from above and they do it for different reasons.

The stack. L layers of paper t thick is L·t of stack, and a fold stops working when that approaches the smallest feature the folder is working to — call it f. So L < f ⁄ t, and thinner paper raises it. That is the ladder’s first rung, in one line.

The grid. To pile L layers over a point, a design needs about L divisions across the sheet. A box-pleated design is laid out on a grid and its deepest pile is of the order of the grid’s fineness, and a division cannot be narrower than the finest crease a pair of hands can place — call that c. So L < S ⁄ c for a sheet of side S, and a larger sheet raises it.

Both are upper bounds, so the usable layer count is the smaller of the two. Which one is smaller is the question.

Where they cross #

Setting the two equal gives

S* = c · f ⁄ t

and below that sheet size the grid is binding while above it the stack is. Nothing else enters.

Take the smallest workable feature as three millimetres and the finest crease a hand will place as one, which are a folder’s working figures rather than constants of nature — the same three millimetres the ladder’s first rung prices a stack against. Then S* is three thousand microns divided by the paper’s thickness, in millimetres:

• Copier paper, a hundred microns: S* is thirty millimetres. Every sheet anybody uses is far above it, so the stack binds always.
• Newsprint, sixty-five microns: forty-six millimetres. The same.
• Washi, forty microns: seventy-five millimetres. The same, at every ordinary sheet.
• Foil-backed tissue, twenty-six microns: a hundred and fifteen millimetres. An A6 sheet is below it.
• Unryu tissue, eighteen microns: a hundred and sixty-seven millimetres. A6 and A5 are both below it.

Thinner paper pushes the crossing up. That is the direction that has to be got right and it is the counter-intuitive one: the better the paper, the larger the range of sheet sizes over which the paper is not the limit.

Which turns the ladder’s first rung over #

The first rung’s argument is that the complex tradition is downstream of thin paper. Nothing here contradicts it and the crossing adds a second half.

On the papers a classical folder had — a hundred microns of ordinary sheet, forty of good washi — the stack is binding at every sheet size worth using, by a wide margin. The substrate really is the limit, and the argument stands exactly as the first rung makes it.

Push the paper to tissue weights and the crossing rises past the size of a sheet somebody would use for a small model. At that point the limit stops being the paper and becomes the folder, and the design’s layer count is bounded by how fine a crease a pair of hands can place rather than by how thin the sheet is.

So the substrate argument has a natural end. Thinning the paper improves a design’s ceiling until the ceiling is no longer the paper’s, and past that thinning it further buys nothing whatever — the hand has not improved.

That is a claim with a date in it that this collection cannot supply. What can be said is where it would be: at the point in a tradition’s history when its finest papers crossed into the regime where its own sheet sizes put the grid below the stack. The end of the substrate era is a crossing, and it is computable from two material numbers and one number about hands.

What the number about hands is #

The crossing depends on c, the finest crease a folder can place, and c is the only quantity here that is not a material property. It deserves scrutiny because everything turns on it.

A millimetre is generous. A folder working on a thirty-second grid of a fifteen-centimetre sheet is placing creases about five millimetres apart, which is easy; on a sixty-fourth grid they are two and a half millimetres, which is careful work; and past that the divisions are hard to place accurately and harder to place repeatably, which matters more because a grid’s errors accumulate along it.

The collection has measured the accumulation elsewhere. An error at a crease is folded too, and a systematic misplacement grows in proportion to the number of creases while a random one grows as the square root — so a grid of many divisions is not merely fiddly, it is a place where a small bias becomes a large one.

That makes c less like a resolution and more like a tolerance budget, and it means the value of c that matters depends on how long the grid is. A finer c on a short grid is achievable and the same c on a long one is not, which pushes the crossing down for large designs — in the same direction as everything else here.

The regime the design field actually works in #

Set the numbers against the practice and the picture is consistent, which is worth noting because it is weak evidence that the model is about the right thing.

Complex designs are folded on large sheets of thin paper, and the era that produced them began when patterns rather than sequences were published. Both of those raise both ceilings — a large sheet raises the grid ceiling and thin paper raises the stack ceiling — so the practice is pushing on both constraints at once, which is what a practice up against two independent bounds would do.

If only the stack bound mattered, sheet size would be irrelevant to a design’s depth and folders would use whatever was convenient. If only the grid bound mattered, paper weight would be irrelevant and folders would use ordinary paper on big sheets. Neither is the case.

The practice is consistent with both bounds being close to binding, which is what a well-adapted craft looks like: pushed to the point where the constraints meet, because relieving either one alone stops helping.

That is a description rather than a measurement, and it is the kind of statement this field has to be careful with. It is offered as a consistency check on the model and not as evidence for it.

The two ceilings do different things to a design #

Being bounded by the stack and being bounded by the grid are not the same predicament, and the difference shows in what a designer would do about it.

Stack-bound means the model has a place where the paper is too thick to fold. The cure is local: move the layers about, thin the region, route a flap elsewhere. It is a problem at a point, it is visible while folding, and it is why complex designs have a characteristic worst spot that everybody who has folded one can name.

Grid-bound means the divisions are too fine to place. The cure is global: use fewer divisions, which changes the design rather than the model, or use a bigger sheet, which changes nothing about the design at all. It is a problem across the whole sheet, it shows up before any folding of consequence has happened, and it is why grid-based work is so sensitive to sheet size.

So the two constraints have different signatures in practice, and the signatures are the sort of thing a practitioner would recognise even without the arithmetic. A design that fails at one place is stack-bound; one that fails everywhere at once is grid-bound.

That gives the crossing a way to be checked against something other than itself. If the arithmetic is right, complaints about paper thickness should be reports of local failures and complaints about sheet size should be reports of global ones — and those are different sentences that a repertoire’s own instructions would carry.

What a bigger sheet actually buys #

The grid ceiling improves with sheet size and it is worth being careful about what “improves” means, because it is not what it sounds like.

A bigger sheet at the same grid gives wider divisions, which are easier to place. A bigger sheet at a finer grid gives the same division width and more layers. So the sheet size buys layer count only if the designer spends it on divisions rather than on a larger finished model.

And the size the model comes out at is fixed: the finished size is the sheet divided by the square root of the mean layer count. Spending the extra sheet on divisions makes the model smaller relative to the paper, because the layer count went up.

So the two rungs of this ladder interact rather than sitting side by side. A designer who buys a larger sheet to get more layers ends up with a model no larger than before, since the square root of the extra layers eats the extra sheet. The two ceilings are ceilings on different things and the resource that relieves one is the resource the other spends.

That is the sharpest form of what the substrate does to a repertoire. Thin paper raises the depth available; large paper raises the depth available and does not raise the size; and a tradition that wants both a deep model and a large one needs the paper to improve in both directions at once.

Why the crossing is a product and not a difference #

The crossing has a shape worth noticing, because it explains why it moves so far for such small changes in the inputs.

S* = c · f ⁄ t is a product of three quantities, so a change in any of them scales the whole crossing. Halve the finest crease a folder can place and the crossing halves. Halve the paper’s thickness and it doubles. Reduce the stack a folder will accept from three millimetres to two and it falls by a third.

A quantity that is a product of three uncertain numbers is a quantity with a wide range, and the range here is real: across the papers compared the crossing runs from thirty millimetres to a hundred and sixty-seven, a factor of five and a half, on thicknesses spanning the same factor.

So the honest statement of where the handover falls is not a number but a formula with three inputs, two of which are material properties nobody disputes and one of which is a folder’s judgement nobody has measured. The uncertainty is concentrated entirely in c, which is the one this collection has no value for, and the essay’s structure follows from that: everything else is arithmetic and the whole answer rests on the quantity that is not.

That is a common position for an argument to be in and it is worth recognising rather than glossing. A result that is exact in its algebra and rests on one unmeasured parameter is a result whose value is that it says which measurement to make.

Which theorem was checked and how #

The crossing is checked rather than solved and drawn. The grid ceiling is evaluated at the computed crossing and must reproduce the stack ceiling there, to a part in a billion, on every paper — which is the algebra checked against the arithmetic that uses it.

The direction is asserted. The thinnest paper must have a higher crossing than the thickest, and the figure refuses if it does not. That is the counter-intuitive half of the finding stated as a condition on the picture existing, and it is the one a sign slip would reverse.

Both regimes must appear. A table in which every cell was limited by the same constraint would show no crossing, and the figure refuses it — which is why the sheet sizes drawn reach down to the small ones.

And the allowed counts must span a real range. A comparison over papers whose ceilings were all similar would be a table with nothing in it, and the range is required to be at least threefold.

Where the model stops #

The grid ceiling’s form is a modelling choice. Taking the deepest pile as of the order of the grid’s divisions is right for box-pleated work and is not a theorem; a design whose depth comes from a different mechanism has a different relation between its grid and its layers.

c and f are a folder’s working figures. Neither is measured and neither is a constant. The essay states the crossing as a formula in them, so a reader who prefers different numbers can carry them through — which is the honest way to use a parameter nobody has measured.

The paper thicknesses are stated and are not measurements taken here. They span the range these materials cover and the essay’s argument is about the shape of the crossing, which is the same shape at any thickness.

And no date is attached to anything. When a tradition’s papers crossed into the hand-limited regime is a documentary question, this collection’s record carries no sheet weights, and inventing one would be the failure this field exists to avoid.

What the picture cannot show #

The table gives a layer count for each pair and cannot show a design. No model is measured, and whether any design ever sat at the ceiling of either constraint is not in it.

Nor can it show the interaction between the two, which is the thing a designer would most want. A design near both ceilings is in a different position from one near either, and the table draws the smaller of two numbers without saying how close the other one is.

The most conspicuous absence is the folder. c is the only quantity in the crossing that belongs to a person rather than to a material, and it is the one the figure cannot compute, cannot check and cannot vary from evidence. Everything about when the substrate stopped being the limit depends on it.

The idealisation, named #

The stack is layers times thickness with nothing between them; the grid is uniform; the crease is a line with a placement error and no width of its own.

The last of those is the one that matters most and it is the one the ladder below this one has already dismantled. A crease has a radius, and a crease with a radius consumes surface — so a fine grid loses paper to its own creases in proportion to the number of divisions, which is a third ceiling this rung has not drawn.

That third ceiling behaves like the grid one: it gets better as the sheet grows relative to the crease. So including it would move the crossing rather than adding a new regime, and it moves it in the direction that makes the hand-limited region larger.

The honest summary is that there are at least three bounds on a design’s depth, two of them belong to the paper and one to the folder, and the point of this rung is that the folder’s is not always the loosest.

Where the ladder goes next #

This ladder has now priced everything it can price from the material and the arithmetic, and what it owes is a measurement of the one quantity that is neither.

How fine a crease can a folder place, repeatably, along a grid of stated length? That is an experiment with a definite answer, it is the parameter the crossing turns on, and this collection has no value for it. It would also be a good experiment, because the answer is presumably a function of the grid length rather than a constant, and the shape of that function decides how the crossing moves for large designs.

Sideways from here, the crossing belongs beside what a grid costs a designer, which prices the same divisions in a different currency. That ladder asks what a grid gives up in packing efficiency; this one asks what a grid costs in placeable creases; and a design is paying both at once.

The habit worth carrying is about arguments from a material. When a material constraint is shown to bind, ask what it would take for it to stop binding — because a constraint that improves without limit while its competitor does not will eventually hand over, and the handover is usually the interesting date.