Generator

The paper had to arrive first — the paper thinness generator

A generator in the who found it, and when library, called 42 times across 9 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

paper-thinness is one function. Everything below was drawn by it when the site was last published, at settings taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

With nothing specified

`view: "substrate", show: "levels"`

`view: "substrate", show: "tolerate"`

`view: "substrate", show: "curve", tol: 0.01`

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

the thinnest paper here reaches 128 layers where the thickest reaches 16 — the layer count is set by the substrate ×5
2 of the 6 layer counts need a sheet larger than any of the 4 offered — the bound is on the page rather than off it ×4
3 of the 42 pairs are limited by the folder's hand and 39 by the paper — both regimes are on the page ×4
the sheet each model needs is its footprint times the square root of its layer count, and the areas balance to 0.0e+0 at every row ×4
the sheet needed grows from 300 mm to 1697 mm across the range — 5.7× for 32× the layers ×4
thinner paper pushes the crossing UP — unryu tissue hands over to the stack only past 167 mm, where copier paper does so at 30 mm ×4
the fibre is 25 microns across, which is stated rather than derived, and every count below is that division ×2
the floor is a fibre and a half of paper, the stack is 3 mm, and the sheet is 1200 mm — three numbers from three separate arguments and no others ×2
the layer count these pairs allow spans 5.6× — from 30 on the thickest paper to 166 on the thinnest ×2
the papers here run from 0.7 fibres through the thickness to 4.0, so a floor at one or two fibres is inside the range rather than outside it ×2
the two ceilings change places at a finished size of 134 millimetres — below it the stack binds and above it the sheet does ×2
the two ceilings meet exactly at S* = cell × feature ⁄ t, reproduced to 0.0e+0 on all 6 papers ×2
6 of 8 printed patterns hold their deepest depth over at least a tenth of the footprint — the preliminary base 99.8%, the miura fold 11.7%, the square twist 17.4%, the hexagon twist 24.4%, the yoshimura pattern 100.0%, the waterbomb tessellation 98.1% — and 2 over less: fold and cut — the triangle 2.91%, the tapered corrugation 0.78% ×1
6 papers are compared, each with a measured thickness ×1
a hundredth of the footprint allowed proud moves 1 of 8 patterns: the tapered corrugation from 16 layers to 12, its corner from 186 to 161 mm ×1
and every paper below a fibre and a half is a tissue — the two the tradition either backs with foil or does not crease at all ×1
and no size anywhere reaches more than 80 layers, because the thinnest paper that holds a crease is 38 microns and its stack runs out there ×1
and the sheet a finished model of 150 mm needs at 4, 16, 64, 128 layers lands inside that bracket at 16 layers and 64 layers and 128 layers, which is why the bound is worth computing at all ×1
and the sheet a finished model of 75 mm needs at 16, 64, 256 layers lands inside that bracket at 64 layers and 256 layers, which is why the bound is worth computing at all ×1
and what is left of it is the corner of the map: the smallest sheet drawn, 74 mm, on a paper at the crease floor ×1
and with it the hand binds in 1 cell of 42 rather than in 6 — the region where the hand is the limit is almost entirely inside the region where no crease holds ×1
every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×1
fold and cut — the triangle: deepest 7, mean 1.19, held at 0.5% proud 7, 1.0% proud 7, 2.0% proud 7, 5.0% proud 1, 10.0% proud 1; corner 326 mm, at 5% proud 123 mm ×1
on every pattern a tolerance changes the depth to be held exactly when it passes the share of the footprint at the deepest depth, and not before ×1
on every printed pattern the deepest point of the folded state is at least the mean depth, as it has to be ×1
over the whole range of frames and arms the largest square sheet runs from 495 to 2122 millimetres — a factor of 4.3, so the bound is bracketed rather than pinned ×1
so the most layers any paper on this map reaches — 75 at 40 microns — is reached with the stack as the limit, which is the thinning argument's claim and not the hand's ×1
the hexagon twist: deepest 7, mean 3.25, held at 0.5% proud 7, 1.0% proud 7, 2.0% proud 7, 5.0% proud 7, 10.0% proud 7; corner 197 mm, at 5% proud 197 mm ×1
the layer map's deepest point on fold and cut — the triangle is the census's, 7 layers ×1
the layer map's deepest point on the hexagon twist is the census's, 7 layers ×1
the layer map's deepest point on the miura fold is the census's, 16 layers ×1
the layer map's deepest point on the preliminary base is the census's, 8 layers ×1
the layer map's deepest point on the square twist is the census's, 9 layers ×1
the layer map's deepest point on the tapered corrugation is the census's, 16 layers ×1
the layer map's deepest point on the waterbomb tessellation is the census's, 32 layers ×1
the layer map's deepest point on the yoshimura pattern is the census's, 60 layers ×1
the lift is arithmetic over a stated frame density and water ratio and derives no material property; what it is for is where the bracket falls ×1
the linear shrink is the square root of the layer count because the folded footprint times the mean layers is the sheet's area, exactly ×1
the miura fold: deepest 16, mean 9.23, held at 0.5% proud 16, 1.0% proud 16, 2.0% proud 16, 5.0% proud 16, 10.0% proud 16; corner 177 mm, at 5% proud 177 mm ×1
the papers span 5.6x in thickness, and that factor is the whole of what changed ×1
the patterns at a ratio of one have nearly all their footprint at the deepest depth, and every pattern above one and a half has under a third of it there ×1
the preliminary base: deepest 8, mean 7.99, held at 0.5% proud 8, 1.0% proud 8, 2.0% proud 8, 5.0% proud 8, 10.0% proud 8; corner 134 mm, at 5% proud 134 mm ×1
the ratio runs from 1.00 to 5.89 across the shelf, and the patterns at one — the yoshimura pattern, the preliminary base, the waterbomb tessellation — are the ones whose every panel lies over every point ×1
the square twist: deepest 9, mean 3.02, held at 0.5% proud 9, 1.0% proud 9, 2.0% proud 9, 5.0% proud 9, 10.0% proud 9; corner 232 mm, at 5% proud 232 mm ×1
the stack ceiling improves with thinner paper and the grid ceiling with a bigger sheet, so which one binds is a property of the pair and of neither alone ×1
the tapered corrugation: deepest 16, mean 8.33, held at 0.5% proud 16, 1.0% proud 12, 2.0% proud 12, 5.0% proud 12, 10.0% proud 12; corner 186 mm, at 5% proud 161 mm ×1
the waterbomb tessellation: deepest 32, mean 31.48, held at 0.5% proud 32, 1.0% proud 32, 2.0% proud 32, 5.0% proud 32, 10.0% proud 32; corner 135 mm, at 5% proud 135 mm ×1
the yoshimura pattern: deepest 60, mean 60.00, held at 0.5% proud 60, 1.0% proud 60, 2.0% proud 60, 5.0% proud 60, 10.0% proud 60; corner 134 mm, at 5% proud 134 mm ×1
without the crease floor both of the earlier bounds bind somewhere — 6 cells grid, 36 cells stack ×1

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

A proud hundredth buys almost nothing

A folded design's sheet is set by its mean layer count and its paper by its deepest point, and the ratio of the two moved the size at which a design uses both equally. That ratio is two numbers of a distribution. Read the whole distribution of every printed pattern's folded footprint and the shelf splits in two: six patterns reach their deepest depth over at least a tenth of the footprint, and two reach it only on a sliver — the fold-and-cut triangle on 2.9 per cent, the tapered corrugation on 0.78. A bound that lets part of the footprint stand proud of the stack's ceiling does nothing at all until the proud share passes the deepest depth's share, so a hundredth moves one pattern of eight, and a twentieth moves two.

A sheet has a size as well

The layer count a design reaches is fixed by how thin the paper is. What size the finished thing comes out at is fixed by how large the sheet is, through a factor the pattern decides: the folded footprint times the 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.

A sheet is as large as two arms

A model's finished size is its sheet divided by the square root of its layer count, so a sixty-four-layer model at a hand's width wants more than a metre of paper. A hand-made sheet is formed on a mould somebody lifts out of a vat and shakes, and what that bounds is an area rather than a thickness: over the whole plausible range of mould weights and what arms can do repeatedly, the largest square sheet runs from about half a metre to about two. The demand and the bound are the same sizes, which is the one thing about them nobody has to know the constants to see.

Closer than a crease is wide

One fold from a bare square leaves nine marks, seventy-five millimetres apart. Two folds leave five hundred and sixty-five, the closest pair half a millimetre apart. Three folds — using one axiom of the seven — leave half a million, and ninety-four per cent of them have another mark within a fifth of a millimetre. What bounds a folder is not what the axioms reach; it is what the paper can tell apart.

Eighty layers and the sheet decides the rest

Five of these essays each bound one thing and none of them bounds a design. Put together they close. The crease floor fixes the thinnest usable paper at about a fibre and a half; that paper's stack runs out at eighty layers; the largest sheet two arms can make falls away as the square of the finished size. The region under both is every model anybody can fold, and it has a ceiling at eighty layers and a corner at about a hand's width — above which the paper is no longer the limit and the vat is.

The deepest point pays for the paper

A folded design uses its sheet according to its mean layer count and its paper according to its deepest point, and the ratio of the two is a property of the crease pattern. Measured on every printed pattern it runs from exactly one to nearly six — and it does not split tessellations from bases, as expected. It splits patterns whose every panel lies over every point from patterns that keep a footprint with structure in it. The ratio moves the corner of the substrate map by its square root, so the fold-and-cut triangle can reach the paper's eighty layers at 326 millimetres where the preliminary base must stop at 134.

The paper had to arrive first

A model with sixty-four layers at its thickest point, folded in ordinary copier paper, is six and a half millimetres of stack. The layer count a design can reach is fixed by the substrate, not by the folder — so the elaborate tradition is downstream of a manufacturing achievement with its own dates.

The paper that will not hold a crease

Every constraint these essays have found improves as the paper gets thinner: the stack, the size, the layer count. A crease does not. A crease is a plastic hinge in the fibres at the fold, and a sheet one fibre thick has nothing through its thickness to hinge — so there is a floor under the thickness that no manufacturing skill moves, because the fibre diameter is a constant of the plant. The papers a tradition folds sit between one and a half fibres and four, and the two below that in this collection's own shelf are tissues, which are backed with foil before anybody creases them.

Which ceiling is binding

Two constraints hold a design's layer count down and both are ceilings on the same number. The stack gets better as the paper thins; the grid gets better as the sheet grows, because piling layers needs divisions and a division cannot be finer than a folder can place it. 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, and only at tissue weights does the hand take over.

All the figures · The who found it, and when field · The patterns a reader can fold