Origami & Folding
what one sheet can do
Concept

Strain — where it appears

How much a length of material has stretched or shrunk, as a share of what it was. Folding asks paper for none, so wherever a shape needs it — round a gathered cap, at a tuck that hides too much or too little — strain is the measure of what the paper is being asked to give.

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

5%10%15%15°30°45°60°75°90°strain the rim must takehow much of a sphere the cap coversdry paper1% of stretchreaches 14°damp paper3% of stretchreaches 24°wet-folded6% of stretchreaches 35°a hemisphere needs 36% and nothing made of cellulose is going to supply it

Paper that stretches on purpose

Wet-folding breaks the assumption every theorem of flat folding rests on, deliberately. It does not repeal the geometry — it buys a few percent of strain, and a few percent of strain is worth about twenty degrees of sphere.

material · Idealisation the worst strain straight tucks leave, against the number of starts placed for straindots: the placement's own worst strain; dashed: I² ⁄ 8m², with I = ∫ √(sin s ⁄ s) ds over the cap-5-4-3-2-1starts, on a doubling scalelog₁₀ of the worst strain2481632the dialled cap: 90°I = 1.464small dots: caps of 30° and 150°every doubling of the starts divides the worst strain by four, on every cap

A start is worth an inverse square

Straight tucks gathering a flat disc into a cap leave a worst strain that falls as the starts multiply, and every essay on them counted starts without saying what one buys. It buys an inverse square. A chord of the hiding curve misses it by its width squared times the curve's bend over eight, so starts placed for strain leave a worst strain of I² ⁄ 8m², with I one integral of the cap's shape — 1.464 for a hemisphere. A paper that gives ε therefore needs about I ⁄ √(8ε) starts, a tenth of the give costs three times the starts, and letting the hiding cross the curve saves exactly a factor of √2. The same argument says why the starts spread outward, and by how much.

material · Developability

Named alongside it

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

ConeDevelopabilityGaussian curvatureGoreOptimisationShape retentionSpherical capWet-folding

All concepts