Origami & Folding
what one sheet can do
Generator

Folded at random, and drawn at random

A generator in the curves and material library, called 20 times across 8 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.

crumple-pattern 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

Folded at random, and drawn at randomLeft, the creases a square is left with after eight folds along randomly chosen lines, unfolded. Right, the same number of creases drawn on an uncreased square at random. The two patterns are equally disorderly and their vertices are nothing alike: every vertex of the folded sheet satisfies the flat-folding condition and almost none of the drawn one does.folded 8 times, then unfolded33 interior vertices, all of degree 433 of 33 satisfy Kawasakithe folding is the reason, not the drawing45 creases drawn at random485 interior vertices, all of degree 40 of 485 satisfy Kawasakisame count, same sheet, nothing folded

folds: 6, seed: 23

Folded at random, and drawn at randomLeft, the creases a square is left with after eight folds along randomly chosen lines, unfolded. Right, the same number of creases drawn on an uncreased square at random. The two patterns are equally disorderly and their vertices are nothing alike: every vertex of the folded sheet satisfies the flat-folding condition and almost none of the drawn one does.folded 6 times, then unfolded24 interior vertices, all of degree 424 of 24 satisfy Kawasakithe folding is the reason, not the drawing34 creases drawn at random341 interior vertices, all of degree 40 of 341 satisfy Kawasakisame count, same sheet, nothing folded

folds: 10, seed: 3

Folded at random, and drawn at randomLeft, the creases a square is left with after eight folds along randomly chosen lines, unfolded. Right, the same number of creases drawn on an uncreased square at random. The two patterns are equally disorderly and their vertices are nothing alike: every vertex of the folded sheet satisfies the flat-folding condition and almost none of the drawn one does.folded 10 times, then unfolded64 interior vertices, all of degree 464 of 64 satisfy Kawasakithe folding is the reason, not the drawing82 creases drawn at random2019 interior vertices, all of degree 40 of 2019 satisfy Kawasakisame count, same sheet, nothing folded

folds: 12, seed: 41, view: "one"

Folded at random, and drawn at randomLeft, the creases a square is left with after eight folds along randomly chosen lines, unfolded. Right, the same number of creases drawn on an uncreased square at random. The two patterns are equally disorderly and their vertices are nothing alike: every vertex of the folded sheet satisfies the flat-folding condition and almost none of the drawn one does.folded 12 times, then unfolded83 interior vertices, all of degree 483 of 83 satisfy Kawasakithe folding is the reason, not the drawing
a sheet folded at random — sheet 150×150 mm — 88 mountain, 103 valley, 2520.77 mm of crease

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 crumple has no tail

The least structured crease pattern this collection can produce is a sheet folded at random and flattened. Its consistent letterings get rarer as it deepens — thirty-four of forty down to eleven — and finding one costs one step per panel from beginning to end, with no wrong guess anywhere. Disorder and difficulty turn out to be unrelated quantities.

Every facet is a layer

Fold a sheet at random as many times as patience allows, then count three things: the creases it carries, the facets they cut it into, and the layers in the stack. The last two are the same number, always, and it is one more than the first — so how deep a crumpled sheet folds can be read off the flattened pattern without folding anything.

Rare is not hard

Crumple a sheet deeper and the share of its labellings that agree with themselves falls from thirty-four in forty to eleven. The number of steps a search needs to find one of them does not move at all: it stays at about one per panel, with no backtracking, the whole way down. How often an answer turns up at random and how much work it takes to find one are different quantities, and a crumpled sheet is where they come apart.

The crease that stops in the middle

A sheet folded flat at random writes a crease pattern that satisfies every condition in the subject, everywhere. Leave one layer behind on each fold — one layer out of a dozen — and it stops writing crease patterns at all: the creases stop in the middle of the paper, and a crease with a loose end is a thing no flat folded sheet can have.

The creases a sheet gives itself

A crease pattern drawn at random satisfies the flat-folding condition at essentially none of its vertices. A sheet crumpled at random satisfies it at every single one, on every seed, at every size — and the reason is a tautology that is very easy to miss.

The crumple keeps its options

Every crease pattern this site prints has exactly one folded state and not one of its thirty-nine available rearrangements is legal. A sheet creased by folding it at random four times has an average of 2.16 folded states, one of them has nine, and six of three hundred and thirty-five rearrangements are legal. The sheet nobody designed is the one with room left in it.

The decision a crumple has taken

A sheet crumpled at random satisfies every condition in the subject, because it just folded. It also wrote itself a lettering — one of very many the pattern admits — and it is now in a piece of that space it cannot leave: at six folds a crumpled sheet carries thirty-nine buried creases, which is half a million million million pieces, and every change it admits stays inside one of them.

The letters a crumple was given

A sheet creased by folding it and folding it again arrives with a mountain-valley labelling that cannot be wrong, because a folding produced it. Nothing about the pattern protects it: reletter the same creases and the share of labellings whose letters agree falls from every one of forty at eight panels to eleven of forty at forty-one. The foldability of a crumple is a fact about its history, not about its drawing.

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