A proud hundredth buys almost nothing

The deepest point pays for the paper split a folded design’s demand on its materials in two. The sheet is consumed by the mean layer count: a folded footprint times its mean depth is the sheet’s area. The paper is stopped by the deepest point: the thinnest paper that holds a crease stacks to eighty layers and no further, as eighty layers and the sheet decides the rest found, and a design reaches that ceiling at its thickest pile, not its average. The ratio of the two, measured on every printed pattern, runs from exactly one to nearly six, and it moves the finished size at which a design uses its sheet and its paper equally by its square root.

It ended by noticing that the ratio is two numbers of a distribution. The natural next measurement is the whole of it — the share of each folded footprint at every depth — from which both uses could be read at once. And it asked a pointed question about the paper’s side. A bound that let a small part of the footprint stand proud of the ceiling — a hundredth, say, a layer or two over — would move each pattern’s corner; the ones whose deepest regions are slivers would move most; and perhaps that tolerance is what folders of thick paper practise without naming it.

The distribution answers the first part and sharpens the second into something stronger than expected. A tolerance moves a pattern’s corner exactly when the tolerance is larger than the share of the footprint at the pattern’s deepest depth, and not at all before. On the printed shelf, a hundredth moves one pattern of eight.

Every depth of every footprint #

Each printed pattern is folded flat and its footprint sampled on a grid of 160 by 160 points; at each point covered, the number of panels over it is its depth. The share of the footprint at each depth is then a histogram, and laying the histogram end to end gives the whole of what the ratio summarised.

The shelf falls into two kinds, and nothing sits between them. Six patterns reach their deepest depth over a broad part of the footprint: the preliminary base over 99.8 per cent and the Yoshimura over all of it, the waterbomb tessellation over 98.1, the hexagon twist over 24.4, the square twist over 17.4 and the Miura over 11.7. Two reach it only on a sliver: the fold-and-cut triangle over 2.9 per cent and the tapered corrugation over 0.78.

The two kinds are not the two kinds the ratio suggested. The ratio put the preliminary base, the Yoshimura and the waterbomb at one, with every panel over every point, and the triangle at nearly six; the Miura and the twists sat between. The distribution puts the Miura and the twists with the uniform patterns: their deepest depth is not the whole footprint, but it is a plateau covering a tenth to a quarter of it, the regions where every column or every pleat overlaps at once. Only two patterns have a deepest point that is genuinely a point.

Why the Miura’s deepest is a plateau #

The Miura is the pattern whose place in the two kinds is least obvious, since its ratio of 1.73 put it among the structured footprints and nowhere near the uniform ones. Its deepest depth, sixteen layers, covers 11.7 per cent of its footprint — a plateau, not a point — and the slant that stacks in one depth says why. A folded Miura’s rows fold exactly onto one another, so every depth is a multiple of the rows; its columns overlap $2/\sin 2a$ deep on average, about 3.1 at the printed slant, and at an odd or fractional count the pile alternates between neighbouring whole numbers all along the strip. The deepest of them, sixteen layers, is reached wherever the most columns overlap, and that is not a region at one end. It recurs at every place along the strip where four columns happen to overlap, and those places add up to a tenth of the footprint however long the strip is.

That is the general reason a tessellation’s deepest depth is a plateau. A pattern that repeats has a pile that repeats, and whatever depth it reaches somewhere it reaches in every period. Three kinds of pile sorted the printed shelf into piles that are the whole footprint, piles confined to a patch and piles deepest on a sliver; the distribution puts numbers on those kinds, and the tessellations fall into the first two because their deepest depth is copied along with everything else. A sliver needs something that happens once — a cut line, the end of a taper, the point of a base.

A tolerance and the depth it must hold #

A stack bound that tolerates a proud share $t$ of the footprint does not ask the paper to hold the deepest depth. It asks it to hold the least depth that is exceeded on no more than $t$ of the footprint, and lets the rest stand proud. With $t = 0$ that is the deepest depth and the earlier bound; as $t$ grows it steps down through the depths, each step taken when $t$ passes the share of the footprint deeper than the depth being held.

Six of the eight columns do not move at all. For the preliminary base, the Yoshimura, the waterbomb, the two twists and the Miura, every tolerance up to a tenth of the footprint leaves the depth to be held at the deepest depth, because the deepest depth covers more than a tenth. A hundredth of the Miura’s footprint allowed proud buys nothing: its sixteen-layer plateau covers eleven of every hundred points.

The tapered corrugation moves at a hundredth, from sixteen layers to twelve. Its sixteen-deep region is 0.78 per cent of the footprint and the next deepest, twelve, covers enough that no tolerance up to a tenth moves it further. The fold-and-cut triangle moves at a twentieth, and it moves all the way: from seven layers to one. Its seven-deep region is 2.9 per cent of the footprint, and the depths between one and seven together cover less than two per cent more, so once the sliver may stand proud the whole of the rest of the triangle is a single layer.

Nothing, then a step #

The shape of every curve is the same: flat at one, then a step down, then flat again until the next depth’s share is passed. A tolerance is worth nothing to a pattern until it is larger than the deepest depth’s share, and then it is worth a whole level of the distribution at once. That is the general statement, and it explains the table without any pattern’s particulars. The earlier essay’s guess that the sliver patterns would move most was right about which patterns; the stronger fact is that the others do not move a little, they do not move at all.

That changes what a tolerance would have to be for the whole shelf to feel it. The hexagon twist’s plateau is 24 per cent of its footprint; a bound would have to let a quarter of the folded object stand proud before the twist gained anything, and by then the tolerance is not a small allowance for a bulge but a statement that the paper need not hold the design. The Yoshimura and the preliminary base never step at all short of a tolerance of nearly the whole footprint: their distributions are a single bar.

What the corners do #

The corner is the finished size where the two limits meet: below it the paper binds and some of the sheet is spare, above it the sheet binds. It sits at the sheet’s size times the square root of the ratio of the depth to be held to the stack’s eighty layers, so lowering the depth to be held lowers the corner and hands more of the range of sizes to the sheet.

With a twentieth allowed proud, the tapered corrugation’s corner moves from 186 millimetres to 161 and the triangle’s from 326 to 123. The other six stay where they were, from the preliminary base’s 134 to the square twist’s 232. The triangle’s move is the extreme the distribution predicts: held to one layer, it behaves as a flat sheet with a small proud wedge, and its paper binds only on models smaller than a preliminary base’s corner. Tolerating its sliver turns the pattern with the largest corner on the shelf into the one with the smallest.

This is also where the tolerance meets the two earlier ceilings. Which ceiling is binding set the stack against the grid and found them crossing at one sheet size; the corner here sets the stack against the sheet and finds them crossing at one finished size. A tolerance acts on the stack alone, and so on both crossings at once: it lowers the stack’s demand without touching what the sheet or the grid can supply. For a plateau pattern it does nothing to either crossing. For a sliver pattern it moves both, and the triangle’s five-per-cent tolerance takes it from the most paper-bound pattern on the shelf to the least.

The move is real in one direction and needs care in the other. A lower corner means the paper stops being the binding limit at a smaller model, which is a gain for anyone who wants to fold the pattern large from thin paper. It does not mean the proud region is free: the proud wedge of the triangle is still seven layers, and on a paper at the crease floor seven layers is well inside the stack’s eighty. The tolerance matters only where the proud part actually passes the ceiling, and the corner is the size at which the held depth, not the proud one, reaches it.

Why only two patterns have slivers #

The two sliver patterns reach their deepest depth for opposite reasons, and the ratio beside the distribution shows it. The triangle is a pattern built to reach its depth at one place. A fold-and-cut pattern folds a sheet so that every edge of the shape lands on one line for a single cut; the layers pile only where the cut goes, and the rest of the sheet lies flat around it. Its depth is concentrated by design, and the star that was cut before it was proved is the same construction for a star.

The tapered corrugation reaches its depth by accident of its taper. Its columns narrow towards both ends, a narrow column piles deeper than a wide one at the same slant, and the deepest pile is where the narrowest columns pile. The slant belongs to the line measures exactly that grading, and shows that leaning each zigzag its own amount evens most of it out; a corrugation graded that way would presumably lose its sliver along with most of its taper’s grading, and with it the only thing a tolerance could buy it.

So the two slivers are a design and a defect, and the tolerance treats them identically. That is a warning about what a single statistic of a distribution can say. The ratio and the proud tolerance both read the deepest depth; neither knows whether the depth was wanted.

What the distribution cannot show #

The depth at a point counts panels, not paper. A panel is a flat face of zero thickness in this model, and a folded state of real paper is thickened, creeped and curved at every crease; a panel is not the unit of depth found that a panel’s thickness cannot even follow a graded pile on the printed Miura. Every share here is a share of a flat, thin footprint.

The grid is 160 by 160 points. A sliver of 0.78 per cent is at most a couple of hundred of the grid’s points, enough to measure and not enough to trust to the second decimal; a finer grid would move the sliver shares slightly and could split a sliver into two depths. The plateau shares are large and stable.

And the shelf is eight patterns. A fold-and-cut pattern and a tapered corrugation are the only slivers on it, and a larger shelf chosen for coverage rather than for what these essays happened to need might hold more — every base with a deep central point, for instance, which the printed preliminary base is not, since it is uniform.

The tolerance, stated #

The model is the one of the whole series. The thinnest paper that holds a crease is a fibre and a half thick, about 38 microns; the stack a folder can close is three millimetres, so eighty layers; the largest sheet two arms can make at the vat is 1,200 millimetres. Every pattern is folded flat and every panel counts one layer at every point beneath it. A tolerance $t$ replaces the deepest depth by the least depth exceeded on no more than $t$ of the footprint.

Whether folders of thick paper work to such a tolerance is not something the record here establishes. The measurement says what such a practice could buy and where: nothing on a plateau pattern until the tolerance is large, and a whole level at once on a sliver pattern. A folder who thins or crushes a small deep region of a thick-paper model is working to a tolerance of this kind, and the distribution says that the practice pays only on designs whose deepest region is small — the tips and points where many flaps meet, rather than the uniform bases and tessellations on this shelf.

How the distributions were checked #

Each pattern’s footprint is read from the same layer map, at the same resolution, that measured its deepest point and mean for the ratio, and the deepest depth each distribution reports is required to equal the census’s for every pattern. The held depth under every tolerance is required to equal the deepest depth exactly when the tolerance is below the share at the deepest depth, and to fall below it as soon as the tolerance passes that share, on every pattern.

Still open: the tips of a real base #

The patterns that would feel a tolerance are the ones not on the shelf. A uniaxial base concentrates its layers at the flap tips and the central point where flaps meet, and its depth distribution is presumably a sliver of great depth over a broad shallow field — the triangle’s shape, reached by design for a different purpose. Measuring the distribution of a real base’s folded footprint, from one of the tree-method layouts, would say whether a tolerance of a hundredth moves it by one level or by many, and whether the corner of a complex model is set by its tips or by its body.

The two uses could also be read from one curve. The sheet’s use is the distribution’s mean and the paper’s is a quantile of it; a design choice that moves the whole distribution — a coarser grid, a thinner pleat — moves both, and whether the two move together or apart is the question a designer choosing between layouts would want answered before folding.

Sideways from here, the paper had to arrive first bounded the stack by the paper’s thickness in the history of the substrate; the distribution says that for most patterns that bound applies to a plateau, not a point, and so to a substantial share of the object a folder is actually holding.

The habit worth carrying is about allowances. An allowance on a small share of a quantity is worth exactly as much as the quantity’s extreme is rare. A hundredth proud buys nothing where the extreme covers a tenth, and a whole level where it covers less than a hundredth; before granting one, read the share at the extreme.