Origami & Folding
what one sheet can do
Concept

Crease density — where it appears

How much crease a sheet carries per unit of area. Measured as length it is a property of the pattern and survives being cut out of a larger one; measured as a count of creases it is not, because a cut divides creases and each half is counted separately.

Named by 5 essays across one field — 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 much a cut adds to a crease count, and to a crease lengthsquare ×150.0% too many12 creases counted for 8 · length 12.675 a unit either waysquare ×225.0% too many40 creases counted for 32 · length 12.675 a unit either waysquare ×316.7% too many84 creases counted for 72 · length 12.675 a unit either waytriangular ×141.7% too many34 creases counted for 24 · length 16.938 a unit either waytriangular ×220.8% too many116 creases counted for 96 · length 16.938 a unit either waytriangular ×313.9% too many246 creases counted for 216 · length 16.938 a unit either wayhexagonal ×141.7% too many34 creases counted for 24 · length 17.691 a unit either wayhexagonal ×220.8% too many116 creases counted for 96 · length 17.691 a unit either wayhexagonal ×313.9% too many246 creases counted for 216 · length 17.691 a unit either wayelongated ×130.0% too many52 creases counted for 40 · length 14.654 a unit either wayelongated ×215.0% too many184 creases counted for 160 · length 14.654 a unit either wayelongated ×310.0% too many396 creases counted for 360 · length 14.654 a unit either wayrhombille ×125.0% too many60 creases counted for 48 · length 23.514 a unit either wayrhombille ×212.5% too many216 creases counted for 192 · length 23.514 a unit either wayrhombille ×38.3% too many468 creases counted for 432 · length 23.514 a unit either waythe length is exact because the two halves of a divided crease add back up

A count is not a length

Cut a rectangle out of a tessellation and it reports fifty per cent more creases than the pattern has, then twenty-five, then seventeen — converging on the truth from above and never reaching it. The crease length per unit area it reports is exact at every size, because the two halves of a divided crease add back up. One measurement survives the cut and the other does not.

material · Crease density four conditions with nowhere to holddevelopabilityholds, at 0 verticesKawasakiholds, at 0 verticesMaekawaholds, at 0 verticesbig-little-bigholds, at 0 vertices6 of 12 of these bands have no flat folded stateand only the panel colouring can see it

A crease with no vertex to belong to

Crease density is measured as length of line per area of paper, and everything else about a crease is measured at the vertex it runs into. A band of paper has creases that run from one edge to the other and meet nothing, so it has density and no vertices at all — and it still refuses to fold.

material · Crease density the length a pattern reports, and the length a sheet of paper hassix of the eight are built on a unit square, so on those the distinction does not arisepatternown widthraw lengthreportedon paperThe Miura fold6.3739.36,6791,049out by 6.37xThe tapered corrugation1.187.81,2431,057out by 1.18xThe preliminary base1.004.8724724the sameThe square twist1.004.7704704the sameThe hexagon twist1.006.1916916the sameThe Yoshimura pattern1.0014.02,3802,380the sameFold and cut — the triangle1.001.7258258the sameThe waterbomb tessellation1.0014.32,2902,290the samea builder working in cells returns a pattern several units across, and not dividing by that is the whole of the error

A length needs a scale

These essays measure crease length, and a crease length is a length in the pattern's own coordinates. Six of the eight printed patterns are built on a unit square, so their coordinates are sheet widths and the distinction never arises. Two are not — a Miura laid out as six cells of unit width spans 6.37 — and on those two the shelf multiplied by the printed size without dividing by the width. The Miura's folding length was reported as 6,679 millimetres and is 1,049, and the same pattern's printable sheet has carried the right number all along.

material · Crease density 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

Named alongside it

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

Crease lengthCrease patternIdealisationMeasurementBoundaryPaper thicknessTessellationBoundary vertexCrease radiusFlat-foldabilityGluingInterior vertex

All concepts