Origami & Folding
what one sheet can do
Concept

Substrate — where it appears

The material a pattern is folded in, taken as a property that constrains the design rather than as a detail of execution. Its thickness caps the layer count a model can reach and its size caps the finished dimension through the pattern's own shrink, and neither cap is anything a folder can work round.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

a fold stops working when the stack reaches 3 mm8 layers16 layers32 layers64 layers128 layersnewsprint65 µm520 µm1.0 mm2.1 mm4.2 mm8.3 mmcopier paper100 µm800 µm1.6 mm3.2 mm6.4 mm12.8 mmkami70 µm560 µm1.1 mm2.2 mm4.5 mm9.0 mmwashi40 µm320 µm640 µm1.3 mm2.6 mm5.1 mmfoil-backed tissue26 µm208 µm416 µm832 µm1.7 mm3.3 mmunryu tissue18 µm144 µm288 µm576 µm1.2 mm2.3 mmthickness measured across the sheet; the smallest feature is a folder's working figurerather than a constant of nature

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.

history · Paper as substrate what the design asks the mill forevery model finished 150 mm acrossA4A2A1A04 layersshrinks 2.00× across300 mm — A2 will do8 layersshrinks 2.83× across424 mm — A2 will do16 layersshrinks 4.00× across600 mm — A1 will do32 layersshrinks 5.66× across849 mm — A0 will do64 layersshrinks 8.00× across1200 mm — larger than any of these128 layersshrinks 11.31× across1697 mm — larger than any of thesea finished model 150 mm across · sheet side = 150 mm × √(mean layers) · the shrink is the square root, so the paper grows quickly

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.

history · Paper as substrate what a paper and a sheet size leave between themthe largest layer count both ceilings allow, and which one decided itpaper105 mm148 mm210 mm297 mm420 mm594 mmhands over atnewsprint65 µm46464646464646 mmcopier paper100 µm30303030303030 mmkami70 µm42424242424243 mmwashi40 µm75757575757575 mmfoil-backed tissue26 µm105115115115115115115 mmunryu tissue18 µm105148166166166166167 mmstack tolerance 3 mm, finest crease 1.0 mm · stack: L < feature ⁄ t · grid: L < sheet ⁄ cell · crossing at cell × feature ⁄ t

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.

history · Paper as substrate the stack under a single cuta clean cut taken as 0.6 mm of stackstarnewsprintcopier paperkamiwashifoil-backed tissueunryu tissue3 points6 layers390 µm600 µm420 µm240 µm156 µm108 µm4 points8 layers520 µm800 µm560 µm320 µm208 µm144 µm5 points10 layers650 µm1.0 mm700 µm400 µm260 µm180 µm6 points12 layers780 µm1.2 mm840 µm480 µm312 µm216 µm8 points16 layers1.0 mm1.6 mm1.1 mm640 µm416 µm288 µm10 points20 layers1.3 mm2.0 mm1.4 mm800 µm520 µm360 µm12 points24 layers1.6 mm2.4 mm1.7 mm960 µm624 µm432 µm16 points32 layers2.1 mm3.2 mm2.2 mm1.3 mm832 µm576 µm20 points40 layers2.6 mm4.0 mm2.8 mm1.6 mm1.0 mm720 µma k-pointed star is folded into 2k wedges, so the scissors pass through 2k layers · a clean single cut is taken here as 0.6 mm of stack

How many wedges the paper allows

A k-pointed folk star is folded into 2k equal wedges and cut once, so the scissors go through 2k thicknesses of the sheet. The geometry is indifferent to k and the paper is not: at a stack a pair of scissors will shear cleanly in one pass, ordinary copier paper takes a three-pointed star and nothing more, and the five-pointed one everybody knows needs washi or thinner.

history · Folklore of folding the sheet a mould can carry, and the sheet a design wantsa finished model 150 mm across needs the square root of its layer count in sheet01e+32e+3481420the mould's own weight, kilograms a square metrethe largest square sheet, millimetres a side4 layers wants 300 mm64 layers wants 1200 mm128 layers wants 1697 mm5 kg10 kg20 kga sheet of 40 grams a square metre carrying 10 times its own mass in water, on a mould of that weight, lifted and shaken repeatedly

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.

history · Paper as substrate how many fibres thick each paper is25 microns to a fibrethe fibre width is stated rather than measured here, and the ordering survives any figure near itcopier paper4.0100 µm · 80 g/m² · 30 layers of stackkami2.870 µm · 60 g/m² · 42 layers of stacknewsprint2.665 µm · 45 g/m² · 46 layers of stackwashi1.640 µm · 30 g/m² · 75 layers of stackfoil-backed tissue1.026 µm · 22 g/m² · 115 layers of stackunryu tissue0.718 µm · 12 g/m² · 166 layers of stacka sheet one fibre thick has nothing through its thickness to hinge, which is why the thinnest here are backed rather than folded

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.

history · Paper as substrate three bounds on a hand-made model, drawn as one regionthe crease floor fixes the paper, the paper fixes the stack, and the sheet falls away as the model growslargest sheet 1200 mm02550755075100150200300450600the model's finished size, millimetreslayers the design may reachthe stack stops at 80 layersthe two change places at 134 mma paper at the crease floor of 38 microns, and a stack that stops at 3 mm

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.

history · Paper as substrate the deepest point of each folded pattern, and its mean depthdeepest pointmean over the footprintThe Yoshimura patterndeepest ÷ mean = 1.00 · 100.0% at the deepestThe preliminary basedeepest ÷ mean = 1.00 · 99.8% at the deepestThe waterbomb tessellationdeepest ÷ mean = 1.02 · 98.1% at the deepestThe Miura folddeepest ÷ mean = 1.73 · 11.7% at the deepestThe tapered corrugationdeepest ÷ mean = 1.92 · 0.8% at the deepestThe hexagon twistdeepest ÷ mean = 2.15 · 24.4% at the deepestThe square twistdeepest ÷ mean = 2.98 · 17.4% at the deepestFold and cut — the triangledeepest ÷ mean = 5.89 · 2.9% at the deepestthe sheet is consumed by the mean and the fold is stopped by the deepest point

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.

history · Paper as substrate

Named alongside it

The objects these essays reach for when they reach for this one.

Layer countIdealisationDesign spacePacking ratioThicknessManufactureTrade-offBox pleatingConservationCrease radiusThe fold-and-cut theoremKirigami

All concepts