Generator

How far a hand travels to fold each printed pattern

A generator in the curves and material library, called 50 times across 11 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

crease-density is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

`view: "rate"`

`view: "scale"`

`view: "shelf"`

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

the densest band on any of these carries 1.9 times its share of the paper's own area ×4
crease length per cell moves by a factor of 1.71 across the sizes measured, while the total moves by 20.5 ×3
the closest approach in units of a cell has settled at 0.925 by 24 cells a side, moving 0.7 per cent from 16 ×2
the density of this family is linear in its cell count to 0.7 per cent over the four sizes measured, which is what licenses solving for the ceiling rather than sweeping to it ×2
the finest member of the family a sheet carries rises as the paper thins — 139 on copier paper, 198 on kami, 345 on washi, 531 on foil-backed tissue ×2
2 of the 8 printed patterns are built in coordinates that are not the sheet's, and 6 are — which is why the two readings agree on most of the shelf and disagree badly on the rest ×1
and it is inside by a factor of 19 on the worst paper, so the shelf is nowhere near the material's limit ×1
and it rises by a factor of 3.8 across the papers, which is the thickness ratio and nothing else ×1
and the worst of them is out by a factor of 6.37 — 6,679 millimetres reported against 1,049 on the paper ×1
compaction is the sheet's area over the footprint it packs into, which is the average layer count and is measured rather than designed ×1
density times closest spacing is above one on 3 of the 6 printed patterns with two creases that do not meet, and reaches 2.02 — so a pattern can carry more crease than parallel lines at its own spacing ×1
every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×1
every printed crease on copier paper is longer than the overlap at its two ends — the tightest, on the tapered corrugation, by a factor of 16 ×1
on every paper the miura can be folded finer than the density bound allows before any two creases come within a crease width — 262 cells against 139 on copier paper, 374 cells against 198 on kami, 655 cells against 345 on washi, 1008 cells against 531 on foil-backed tissue ×1
on every paper the miura's first crease to become overlap from end to end arrives at a coarser pattern than its first pair of non-meeting creases does — 135 cells against 262 on copier paper, 193 cells against 374 on kami, 337 cells against 655 on washi, 519 cells against 1008 on foil-backed tissue ×1
on every paper the waterbomb can be folded finer than the density bound allows before any two creases come within a crease width — 141 cells against 74 on copier paper, 202 cells against 105 on kami, 354 cells against 185 on washi, 544 cells against 284 on foil-backed tissue ×1
on every paper the waterbomb's first crease to become overlap from end to end arrives at a coarser pattern than its first pair of non-meeting creases does — 82 cells against 141 on copier paper, 118 cells against 202 on kami, 207 cells against 354 on washi, 319 cells against 544 on foil-backed tissue ×1
ordering the printed patterns by crease count and by folding length gives two different orders ×1
the ceiling is one division — a crease 6 thicknesses wide, laid at its own width — and the densities are the same length-over-area measurement the shelf is priced in ×1
the densest pattern on the shelf asks for 89 metres of crease a square metre and the thickest paper here allows 1667, so every printed pattern is inside every paper's ceiling ×1
the folding length is the sum of the drawn crease lengths, taken off the pattern's own coordinates and multiplied by the size the sheet is printed at ×1
the folding length per layer of compaction varies by 8 times across the printed patterns ×1
the sheet is cut into 5 bands by distance from its own edge, and each band's share of the paper is computed rather than assumed equal — 36%, 28%, 20%, 12%, 4% from the rim inward ×1
the sheet is cut into 6 bands by distance from its own edge, and each band's share of the paper is computed rather than assumed equal — 31%, 25%, 19%, 14%, 8%, 3% from the rim inward ×1
the sheet is the same unit square at every size, so the length is a density and the cells shrink as they multiply ×1

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

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.

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.

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.

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.

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.

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.

The paper a pattern asks for

A Miura of c columns and r rows at a slant α wants a sheet whose proportion is (c + tan α) / r — one equation tying the two counts, the angle and the shape of the paper. A square is the case where it comes to one, which needs the tangent of the slant to be a whole number: 45° for a pattern one row taller than it is wide, 63.43° for two, and nothing at all for the slants anybody draws.

The pattern cheapest to trust

Demonstrating that a one-shot deployment will open takes a number of successful tests proportional to its hinge count, so the pattern that needs fewest tests for what it delivers is the one with fewest hinges per layer of compaction. That criterion is a count nobody computes, and computed on the printed shelf it ranks the patterns differently from crease length per layer: the preliminary base is first, at exactly one hinge per layer, and the square twist rises from seventh to fourth. As patterns are refined the difference sharpens. The waterbomb settles at 2.47 hinges a layer and the Yoshimura at 1.47, but the Miura climbs without levelling — 1.32 at two cells a side, 5.69 at eight — so every finer Miura costs more tests for each layer it adds, and the pattern that gets built is the only one of the three that gets dearer to trust as it gets finer.

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.

Fourth of eight, and still not chosen for it

A deployable is sold on compaction: large in use, small in transit. Measured, the pattern that actually gets built converts folding into compaction at 0.67 sheet-widths of crease per layer, which is fourth of the eight printed patterns — nearly three times worse than the Yoshimura, which nobody deploys, and nearly three times better than the hexagon twist, which nobody deploys either. The ranking does not pick out the pattern that flew from anywhere on the shelf, and that is the finding.

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.

Every generator · The curves and material field · The patterns a reader can fold