Generator

How far a hand travels to fold each printed pattern

A generator in the curves and material library, called 69 times across 14 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

crease-density is one function. Everything below was drawn by it when the site was last published, at settings 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.

With nothing specified

`view: "rate"`

`view: "depth", show: "vertex"`

`view: "depth", show: "types"`

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

the densest band on any of these carries 1.9 times its share of the paper's own area ×4
an evenly spread vertex of degree 4 creases (d/4)·csc(2π/d) = 1.0000 band widths squared at least twice, by the kites and by the polar integral alike ×3
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
3000 flat-foldable degree-six vertices drawn at random all crease at least the even vertex's 1.7321; the least is 1.7363 ×1
along equal sectors the derivative of f in the mean is cos x (cos x − 4)⁄sin²x, negative all the way to a right angle, checked against a difference quotient at 79 points ×1
and at every mean the derivative in the half-difference, 4 sin x sin y (cos²x + cos²y)⁄(sin²x − sin²y)² − tan²y, is positive on the whole grid tried, matching a difference quotient to 3.3e-9 ×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 none of them creases less than one band width squared twice — the least is 1.0000 ×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
at 207 cells a side, where a crease first becomes band overlap from end to end on copier paper, 57.5 per cent of the sheet is creased at least twice ×1
at a mean sector of 70° the doubled ground only rises as the two free sectors part ×1
at a Miura vertex, slant 0.35 the ground creased 4 times over is a disc half a band across, 0.786 band widths squared against π/4 ×1
at degree four no vertex ever has more than two neighbouring pairs under a half-turn, which is what reduces its overlaps to two terms ×1
at degree six, 3000 of 3000 random flat-foldable vertices have more than two neighbouring pairs of sectors under a half-turn and 2397 have all six, so the pairing that closes the argument at degree four is not available ×1
at each paper's own finest Miura the share of the sheet creased at least twice is the same, 57.5, 57.5, 57.6, 57.6 per cent, because the bound fixes the band's width against a cell ×1
at the preliminary base's centre the ground creased 8 times over is a disc half a band across, 0.787 band widths squared against π/4 ×1
at the size each is printed, every pattern on the shelf creases less than one per cent of its sheet twice — the most, the yoshimura pattern, 0.07 per cent ×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 26 ×1
for every vertex tabulated the polar integral confirms the closed form, the vertex creases at least as much as the equal-sector vertex with its mean, and that at least one ×1
none of 400 flat-foldable vertices of degree 6 creases less than the evenly spread one's 1.732 band widths squared twice; the least of them is 1.744 ×1
on 4000 flat-foldable degree-four vertices drawn at random the closed form equals f(x, y)⁄4 to 2.8e-14, and on the first 24 the polar integral agrees to 0.0000 ×1
on a Miura-type vertex the doubled ground falls all the way to the square grid's right angles, where it is exactly one band width squared ×1
on all 8 kinds of vertex on the printed patterns the kites-less-overlaps formula and the polar integral agree to the fourth decimal ×1
on every one of the 8 kinds of vertex on the printed patterns the ground creased as deep as the degree is the same disc, π/4 of a band width squared, whatever the angles ×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 — 207 cells against 262 on copier paper, 296 cells against 374 on kami, 519 cells against 655 on washi, 799 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 — 117 cells against 141 on copier paper, 167 cells against 202 on kami, 293 cells against 354 on washi, 451 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 sampled table was within 1.3 per cent of the closed form on every kind, the spread being the sampling grid's ×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
two creases at 70° share a kite outside the disc of h²(cot(θ/2) − (π − θ)/2) = 0.468 half-bands squared, the same area a polar integral of the doubled ground gives ×1
while the bands are thin against a cell the share creased twice or more is a fixed multiple of the square of their ratio — 1.11, 0.99, 1.12 — because it is a fixed area at every vertex ×1

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

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 disc as deep as the degree

A crease occupies a band a few sheet thicknesses wide, and near a vertex every band contains the point. So the paper there is creased as many times as the vertex has creases — over a disc exactly half a band across, π/4 of a band width squared, at every flat-foldable vertex whatever its angles. Around the disc lies a fringe creased twice, which is what a narrow sector buys: 0.99 band widths squared at a right-angled vertex, 2.86 at the preliminary base's centre. At printed size the doubled ground is a few square millimetres. It is a fixed area per vertex, so it grows as the square of the fineness, and at the finest Miura any paper carries more than half the sheet is creased at least twice.

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⁄2)·cot(θ⁄2) for a sector angle θ below a right angle, and half a band above it. 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 207 cells a side by its vertices against 262 by its spacing, and the finest waterbomb 117 against 141. Both allow half as much again as the density bound, which had the wrong argument and the wrong 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 floor at degree four

A flat-foldable vertex creases some of the paper round it twice, and the ground it doubles had been found by search never to fall below the evenly spread vertex's — one band width squared at degree four. At degree four it can be proved. Kawasaki's condition leaves two free sectors; two identities of the kite turn the closed form into a function of their mean and half their difference; and two derivatives, each one line, show that parting the sectors always costs ground and that, with them equal, opening the mean to a right angle always saves it. So every flat-foldable degree-four vertex creases at least one band width squared twice, and only four right angles crease exactly that. The same pairing that makes the proof work is missing at degree six.

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.

Where two bands part

Two creases leaving a vertex share a kite of paper beyond the disc every band contains: its sides are the bands' far edges, tangent to the disc, and its tip sits on the bisector. The kite is h²(cot(θ/2) − (π − θ)/2), and a whole vertex is its kites less the overlaps of neighbouring kites — exactly 1, √3 and 2√2 band widths squared for the square grid, the Yoshimura and the preliminary base. The same kite says where the bands part along each crease, (w/2)·cot(θ/2), which is shorter than the w/sin θ these essays had charged: the finest Miura on copier paper is 207 cells a side, not 135, and at that size 57.5 per cent of the sheet is creased twice.

All the figures · The curves and material field · The patterns a reader can fold