Origami & Folding
what one sheet can do
Series

Crease density — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. the bar is millimetres of crease at the printed sizea pattern with more creases is not always a pattern with more folding in itThe preliminary base724 mm8 creases · printed at 150 mmThe Miura fold1,049 mm38 creases · printed at 170 mmThe square twist704 mm12 creases · printed at 150 mmThe hexagon twist916 mm18 creases · printed at 150 mmThe Yoshimura pattern2,380 mm86 creases · printed at 170 mmFold and cut — the triangle258 mm6 creases · printed at 150 mmThe tapered corrugation1,057 mm45 creases · printed at 160 mmThe waterbomb tessellation2,290 mm76 creases · printed at 160 mmsix point seven metres of crease on a sheet seventeen centimetres across

    How much line is on the paper

    A crease pattern is described by its creases: how many, at what angles, in what arrangement. What a folder spends is length. The Yoshimura this site prints has eighty-six creases and 2,380 millimetres of folding on a sheet seventeen centimetres across; the Miura has thirty-eight creases and 1,049, and the fold-and-cut triangle has six and 258 — and the two counts do not rank the eight printed patterns the same way.

    part 1 · material
  2. each number is the folding in that band against the paper in it — one is an even spreadthe rimband 2band 3band 4the middleThe preliminary base0.550.720.991.694.96The Miura fold0.431.410.751.941.67The square twist0.660.881.182.030.94The hexagon twist0.600.871.242.440The Yoshimura pattern0.791.011.091.211.77Fold and cut — the triangle00.201.313.446.79The tapered corrugation0.611.241.390.731.63The waterbomb tessellation0.890.931.260.951.34bands are equal in depth and not in area: 36% · 28% · 20% · 12% · 4% of the sheet, from the rim inward

    Where the length sits

    A pattern's folding length is a total, and a total says nothing about where the work is. Divide each printed sheet into bands by distance from its own edge and the answer separates the patterns by kind: a traditional base carries five times its share of folding in the middle 4% of the paper, a tessellation carries between 0.9 and 1.4 everywhere, and a twist unit carries none at all at its centre. On all eight, the outermost band carries less than its share.

    part 2 · material
  3. 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.

    part 3 · material
  4. 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.

    part 4 · material
  5. 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.

    part 5 · material
  6. 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.

    part 6 · material
  7. 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.

    part 7 · material
  8. crease density, the closest two creases come without meeting, and the two multipliedthe closest approach is between creases with no vertex in common, measured as segmentspatternm a m²closest mmproductThe Yoshimura pattern8224.52.02The waterbomb tessellation8920.01.79The square twist3136.11.13The Miura fold3625.10.91The hexagon twist4122.10.90The tapered corrugation4118.60.77parallel creases give a product of exactly one; a pattern above one carries more length than its own spacing would suggest

    A paper limits spacing, not density

    The density ceiling put a sheet's limit at parallel creases a crease-width apart, 1⁄w of crease a square metre, and claimed no arrangement carries more. Crossing families do: they meet at vertices, which is shared ground, and never come closer than w anywhere else. Measured over every pair of creases that do not share a vertex, density times closest spacing settles at about 1.9 for both the Miura and the waterbomb, so each can be folded nearly twice as fine as the density bound said — 262 cells a side on copier paper for the Miura rather than 139, and 141 for the waterbomb rather than 74.

    part 8 · material
  9. the crease each pattern has least room on, once the ground near its ends is creased twicea narrow sector pushes the overlap out along both creases, as one over the sine of the anglepatternnarrowest sectorreach, both endsthat crease, mmroom to shrinkThe tapered corrugation65.9°2.1020.416×The waterbomb tessellation45.0°2.4128.320×The hexagon twist60.0°2.1525.520×The Yoshimura pattern60.0°2.3128.320×The Miura fold69.9°2.0626.722×The square twist90.0°2.0036.130×Fold and cut — the triangle58.2°1.1828.640×The preliminary base45.0°1.4175.088×reach is in band widths; room to shrink is the crease's length over that reach, on copier paper with a band 0.6 mm wide

    A vertex creases the paper twice

    Two creases that meet share ground near the point, and their bands overlap out along each of them to w⁄sin θ for a sector angle θ below a right angle. Add the overlap at both ends of a crease, and a crease no longer than that is overlap from end to end — a crease only in the drawing. That third bound binds before the spacing does: the finest Miura on copier paper is 135 cells a side by its vertices against 262 by its spacing, and the finest waterbomb 82 against 141. Both land within a tenth of the density bound, which had the wrong argument and nearly the right number.

    part 9 · material

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