Generator

A boundary vertex is a strip

A generator in the flat-folding library, called 42 times across 8 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.

edge-vertex 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: "released", pattern: "square-twist"`

`view: "fan", bearings: [40, 100, 120], letters: "MVM", spanDeg: 180`

`view: "shelf"`

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.

across 1296 fans of three creases a full turn wide, with equal end sectors, a fan admits four of its eight letterings exactly when a middle sector is strictly smaller than both its neighbours, six when the two middles are equal and smaller than the ends, and all eight otherwise ×1
all 128 arrays of slits on a 4 by 3 Miura admit exactly the product of their vertices' own shares; the slits' own factors multiply correctly on exactly the 22 arrays in which no two slits meet ×1
and on the diagonal, where the two middle sectors are equal and smaller than the ends, a fan admits six — the two rolls are the only letterings refused ×1
at every one of the 298 released vertices, the letterings the strip refuses are exactly those that fold three neighbouring creases alike round two equal sectors, each shorter than the sector beyond it — one rule for the degree-four and the degree-six tips alike ×1
at rows 1.732 half-columns tall every vertex correlates its pairs by -0.0667, the 27 triangles carry the leading loops, and the count falls short of the product: 0.992525 ×1
at rows 2.200 half-columns tall each vertex only locks pairs, no loop closes through locked pairs, and the count is exactly the product ×1
each of the 12 creases is cut in turn and the whole pattern re-counted, so the comparison is over every cut rather than over a chosen one ×1
each vertex is asked twice — as a ring and as the same sectors in a line — so the two counts are of one object arranged two ways ×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 pattern on the printed shelf is counted, so the comparison is over all 8 of them rather than over the ones that make the point ×1
from rows 1.2 half-columns tall up to the equilateral √3 the count is 0.992525 of the product at every proportion tried, and above √3 it is exactly the product ×1
of the 298 vertices that a single slit through a buried crease releases on the printed shelf, 81 still refuse some of their letterings, on 4 of the 6 patterns that have a buried crease; on the square twist and the Yoshimura pattern none does ×1
on 4 of the 5 printed patterns the share of letterings passing every local test is exactly the product of the vertices' own shares, and on The Yoshimura pattern it is less ×1
on a 4 by 3 Miura, of the 21 pairs of slits the slits' own factors multiply correctly for exactly the 11 pairs that do not meet; every vertex's own share multiplies correctly for all 21 ×1
on a 4 by 3 Miura, which admits 2048 of its 2^17 letterings uncut, every slit through a buried crease multiplies the share the four conditions admit by four, as the ledger says — and with the strip's condition at its two tips, 4 of the 7 slits multiply it by three ×1
on a 6 by 5 Miura every zigzag slit laid apart from the others multiplies the share by four and every straight one by three; zigzag slits laid end to end release all 20 interior vertices with 15 slits, and laid apart with 10 ×1
on a 6 by 5 Yoshimura the loop sum is the counted share over the product, 0.992525, to twelve places; no loop of one or two creases contributes, and the loops of three are its 27 small triangles at (−1⁄15)³ each ×1
on every size tried the shortfall is 4.00e-4 to 4.07e-4 of a power of two per small triangle, just under the 4.28e-4 one triangle alone would cost ×1
over 234 vertices, cutting the ring open never cost a lettering ×1
the fan stops admitting every lettering at 50 of the 61 positions swept, and at every one of them a sector is strictly smaller than both its neighbours ×1
the fan's creases are given in order round the vertex, so the sectors between them are the sectors of the paper ×1
the Miura vertex correlates no pair of its creases, only all four at once; the equilateral Yoshimura vertex correlates every one of its fifteen pairs by −1⁄15; the taller one locks three pairs and leaves the other twelve uncorrelated ×1
the pair differs in one letter, both members pass every condition this site checks, and one of them cannot be folded ×1
the zigzag lines array of 10 slits on a 6 by 4 Miura multiplies the share of letterings that pass every local test by 32768, which is the product of what each vertex it cuts now admits against the half an uncut vertex admits ×1
the zigzag pairs array of 10 slits on a 6 by 5 Miura multiplies the share of letterings that pass every local test by 1048576, which is the product of what each vertex it cuts now admits against the half an uncut vertex admits ×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 cut is a licence

What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried.

A ring and a line

A vertex has a certain amount of paper at it, and the paper either closes round or it does not. Holding the sectors fixed and changing only that: the ring has twice as many letterings to choose from and folds in a quarter of them, the line has half as many and folds in seven-tenths, and cutting a ring open has never once cost a lettering.

A slit tip keeps one condition

A cut was priced at a factor of two in the letterings a pattern admits for every interior vertex it releases — exactly, no more and not less — and the price was measured on the square twist. A released vertex is not free. It is a strip a full turn long, and a strip has its own big-little-big lemma: a middle sector smaller than both its neighbours forces the creases either side of it to differ. Across the printed shelf, 81 of the 298 vertices a single slit releases still refuse some of their letterings. On a Miura every slit along a straight crease multiplies the share by three where the ledger says four, and every slit along a zigzag by four. The square twist, on which the price was set, is one of the two patterns where the strip never bites.

An array is priced by its vertices

One slit along a Miura's zigzag crease multiplies the share of letterings that pass every local test by four, and one along a straight crease by three. An array of slits is not the product of those factors. Two slits that meet at a vertex release it once, not twice, and on a small Miura the slits' own factors price correctly exactly the 22 of its 128 arrays in which no two slits touch. What prices every array is the product of each vertex's own share — the ring conditions where nothing was cut, the strip rule wedge by wedge where something was. So the cheapest way to release every vertex is a matching: zigzag slits laid in pairs that never meet, half as many slits as vertices. And the count can be made before anything is cut only on a pattern whose vertices decide independently, which the Yoshimura's do not.

Most of a patch is edge

Between 34% and 91% of the vertices in the crease patterns drawn here sit on the edge of the paper rather than inside it, and on the tessellation patches — the figures that are meant to show what a repeating pattern looks like — it never falls below a third. A boundary is one unit deep whatever the unit is, so the share falls like one over the number of units across and reaches nothing at any size a page can carry.

The rim lies over less

A folded sheet's boundary is usually discussed as the place the theorems stop applying. It is also visible in the pile: a panel carrying a raw edge of the paper lies over fewer of the other panels than one that does not, on every printed pattern that has both kinds — 18.0 against 21.0 on a Miura, 31.0 against 37.7 on a waterbomb tessellation, and never once the other way round.

The vertices nobody checks

The four conditions of flat folding are stated at an interior vertex, and the word interior carries the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, none of the four conditions says anything about them, and the condition that does decide them — a one-dimensional crease pattern, folded or refused exactly — was in hand all along.

What a loop of creases carries

Multiplying each vertex's own share of letterings gives the exact share for a whole Miura, both twists and a tapered corrugation, and three quarters of a per cent too much for the Yoshimura. The difference has an exact form. The count is the product times a sum over loops in the graph of creases, each loop weighted by the correlations its vertices put on the pairs of creases it uses. A Miura vertex correlates no pair at all, so no loop can carry anything. An equilateral Yoshimura vertex correlates every pair by −1⁄15, and its 27 small triangles take the product down by 0.8 per cent before the larger loops pay a little back. Draw the rows taller than √3 and the big-little-big lemma locks three pairs and frees the rest, no loop closes, and the product is exact.

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