Origami & Folding
what one sheet can do
Generator

Four ways of making a crease pattern, four answers

A generator in the what it costs to know library, called 28 times across 9 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.

pattern-population 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

Four ways of making a crease pattern, four answersFour constructions that each produce crease patterns satisfying every vertex condition, asked the same four questions: what share of each pattern's vertices lie on the edge of the paper, how many layers deep the folded state gets at its worst point, how many times smaller the folded footprint is than the sheet, and how much crease length each unit of paper carries. They are not four samples of one population — no two of them produce the same patterns — and they disagree by factors rather than by margins.each row is an exhaustive count over the patterns that construction producedon the edgedeepest piletimes smallercrease densitythe printed patterns8 patterns62%19.415.5×7.9twist tessellations12 patterns52%9.82.7×12.9quadrilateral meshes6 patterns67%8.74.8×5.2fold-and-cut patterns7 patterns86%10.41.2×2.2

view: "ladder", extra: "patches"

Which refusal fires firstFive ways of saying no to a crease pattern, in order of what they cost, with every member of the four test populations recorded against the first one that refuses it. The cheapest test catches the most, the two in the middle catch nothing here because the cheapest had already caught their cases, and the most expensive is the only one that reaches the rest.the bar is how many of the 38 patterns each refusal is the first to catchtwo creases cross5one sweep over pairs of creasesa vertex condition fails0one pass over the verticesthe panels do not place0one walk over the panelsthe letters force a loop1one pass over the crease listno ordering exists6every ordering of the panels26 of the 38 are refused by none of these and are folded, undecided, or waiting on a search too large to run

view: "ladder"

Which refusal fires firstFive ways of saying no to a crease pattern, in order of what they cost, with every member of the four test populations recorded against the first one that refuses it. The cheapest test catches the most, the two in the middle catch nothing here because the cheapest had already caught their cases, and the most expensive is the only one that reaches the rest.the bar is how many of the 33 patterns each refusal is the first to catchtwo creases cross5one sweep over pairs of creasesa vertex condition fails0one pass over the verticesthe panels do not place0one walk over the panelsthe letters force a loop0one pass over the crease listno ordering exists6every ordering of the panels22 of the 33 are refused by none of these and are folded, undecided, or waiting on a search too large to run

view: "redraw", draws: 60

The populations, with their letters drawn againEach member of each test population relettered independently, and the share of redraws whose letters do not contradict themselves. As the populations stand every member is consistent, which is a fact about the constructions that produced them rather than about the patterns.the bar is the mean share of redraws that agree with themselvesas the populations stand, every member is consistent and the refusal fires on none of themthe printed patterns96.7%8 of 8 could be asked · worst member 90%twist tessellations55.0%7 of 12 could be asked · worst member 7%quadrilateral meshes96.9%6 of 6 could be asked · worst member 82%fold-and-cut patterns100.0%7 of 7 could be asked · worst member 100%a member with no folded state has no letters to redraw and is counted as not asked rather than as passing

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.

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 population nobody chose

Five crease patterns were measured over and over because somebody had drawn five. Ninety-six drawn from a stated grid of tiling, turn and pleat width say something the five could not: nine of them have no consistent lettering at all, and the phenomenon the collection had spent so long measuring belongs to the one tiling the grid leaves out.

A population that cannot fail

Thirty-three crease patterns are kept here to run the checkers over, and every one of them has letters that agree with themselves. That is not a property of the patterns. It is a property of how they were made: each came from a construction that returns a lettering, so a test looking for letters that contradict themselves has nothing to fire on. Reletter the same thirty-three and the failure is available at once — on one member, four of sixty redraws.

Drawn by the same hand

Two straight segments dropped on a square cross about 23% of the time; four of them cross 74% of the time; twelve cross with certainty, about fifteen times over. Every crease pattern in this collection's four test populations has none — not because the checkers were catching them, but because the same rules that drew the patterns were incapable of producing one, and nothing looked until a construction finally did.

Four ways to draw a pattern

Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.

One population, four sheets

A population of patterns is a way of asking what is typical, and it has always been a population of drawings. Put the same drawings on four different sheets and the verdicts move — not because the drawings changed but because the sheet did, which means a population has two halves and only one of them was ever chosen.

The cost is in the coincidences

How big an instance is, is what a hardness statement is about, and it is the weaker predictor of what deciding one costs. Hold the degree fixed and vary only how many of a vertex's sectors are equal: the work of deciding it rises by a factor of nearly three, against a factor of two for doubling the number of creases. The expensive instances are the ones a designer draws on a grid.

The order the refusals come in

This collection can say no to a crease pattern in five ways, and they cost wildly different amounts: a sweep over pairs of creases, a pass over the vertices, a walk over the panels, a pass over the crease list, and an enumeration of every ordering of the panels. Run all five over the thirty-three patterns in the four test populations and the cheapest refuses five, the most expensive refuses six, and the three in between refuse nothing at all.

The patterns a checker is tested on

This site keeps four populations of crease patterns and runs its checkers over them, which is what makes a claim about typical instances measurable rather than rhetorical. Asked whether the members actually fold, the populations answer: thirteen of thirty-three do, six place and cannot be ordered, five cannot be placed at all, and nine are past what the search will finish.

The refusal that reads the list once

There are five ways of saying no to a crease pattern here, and their costs are two hundred and eighty-two, a hundred and twenty-six, a hundred and fifty-seven, thirty-nine thousand six hundred and twenty-one — and a search that is refused outright. On the largest patch the four cheap tests together do less work than one of them looks like it should, and the fifth cannot be started. A refusal that reads the crease list once is the only kind that scales.

Every generator · The what it costs to know field · The patterns a reader can fold