Origami & Folding
what one sheet can do
Concept

Manufacture — where it appears

How an object is actually produced, as distinct from how it is specified. A crease pattern is a drawing and making one is a process with a length, a tolerance and a machine — so a pattern that is geometrically sound can be unmakeable, and a bound on the process is a bound on the design.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

the solved mesh, cut wrong byworst mismatch left, radians0.017 mm0.00210.051 mm0.00580.169 mm0.01990.508 mmno closure at alla Miura, cut wrong by0 per cent2e-141 per cent6e-155 per cent4e-15

Solved is not built

A mesh that folds because an equation holds and a mesh that folds because one crease family runs straight through every vertex are not two examples of the same thing. Cut a Miura's every dimension five per cent wrong and it still folds exactly. Cut a solved general mesh a fifth of a millimetre wrong on a 150 mm sheet and the closure is gone.

rigid · Tolerance 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 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.

Design spaceLayer countPacking ratioSubstrateTrade-offIdealisationMiuraQuadrilateral meshRigid foldingRobustnessTolerance

All concepts