Origami & Folding
what one sheet can do
Concept

Paper thickness — where it appears

How thick a sheet is, measured across it. It decides two things a crease pattern never mentions — how deep a stack of layers becomes, and how wide a crease has to be — so it is the property the idealisation of a sheet as a surface leaves out, and most material constraints on folding are it in disguise.

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

each mark is one crease, ranked shortest to longest10⁻⁵10⁻⁴0.0010.010.1a factor of 498, and no crease in itlength, as a fraction of the sheet's side142 creases, rankedthe 12 in magenta are drawn, counted, lettered and put through every theorem, and none of them is visible

The shortest crease is not a crease

A crease pattern's density is usually quoted as total crease length over sheet area, which treats a metre of folding as a metre whether it arrives as one long line or ten thousand short ones. Reading the lengths individually instead finds twelve creases on a printed patch that are shorter than a wavelength of light.

material · Crease density 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 metres of crease a square metre: what each pattern asks for, and what each paper allowsthe pale bars are patterns and the dark ones are paperswhat foil-backed tissue allows641026 µm, a crease 0.16 mm acrosswhat washi allows416740 µm, a crease 0.24 mm acrosswhat kami allows238170 µm, a crease 0.42 mm acrosswhat copier paper allows1667100 µm, a crease 0.60 mm acrossThe waterbomb tessellation89printed at 160 mmThe Yoshimura pattern82printed at 170 mmThe tapered corrugation41printed at 160 mmThe hexagon twist41printed at 150 mmThe Miura fold36printed at 170 mmThe preliminary base32printed at 150 mmThe square twist31printed at 150 mmFold and cut — the triangle11printed at 150 mma crease occupies about 6 sheet thicknesses, so the closest two creases can be laid is that, and the ceiling is its reciprocal

The density a paper allows

Every density these essays measure is a quotient a pattern hands over, and nothing has asked what the paper's own answer is. It has one: a crease occupies a band a few thicknesses across, so two creases closer than that are not two creases, and a sheet of a given thickness carries a largest density. Copier paper allows 1,667 metres of crease a square metre and the densest pattern on the printed shelf asks for 89 — a factor of nineteen below the worst paper's ceiling. The material is not what limits a crease pattern's density at any fineness anybody folds.

material · Crease density the share of each folded footprint at every depth, shallowest firsteach block is one depth, its length the share of the footprint that deep; the shading cycles so neighbours differThe preliminary base8 deep on 99.8%The Miura fold16 deep on 11.7%The square twist9 deep on 17.4%The hexagon twist7 deep on 24.4%The Yoshimura pattern60 deep on 100.0%Fold and cut — the triangle7 deep on 2.91%The tapered corrugation16 deep on 0.78%The waterbomb tessellation32 deep on 98.1%the tick marks where the deepest depth begins

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.

history · Paper as substrate

Named alongside it

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

IdealisationCrease densityCrease radiusLayer countMeasurementSubstrateBoundary vertexCrease patternDistributionFootprintShape retentionTessellation

All concepts