Origami & Folding
what one sheet can do
Generator

Folding adds and never subtracts

A generator in the curves and material library, called 8 times across 3 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.

crumple-growth 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

Folding adds and never subtractsInterior vertices, facets and total crease length against the number of random folds, averaged over five seeds. Each curve rises and none of them turns over: a fold can only add creases, so the pattern gets finer at every step and the facets between the creases get smaller. Every sheet in the sweep passes the flat-folding condition at every vertex.24681012foldsinterior verticesfacetscrease lengththe median facet falls from 2.6e-1 to 4.0e-3 of the sheet

upTo: 12

Folding adds and never subtractsInterior vertices, facets and total crease length against the number of random folds, averaged over five seeds. Each curve rises and none of them turns over: a fold can only add creases, so the pattern gets finer at every step and the facets between the creases get smaller. Every sheet in the sweep passes the flat-folding condition at every vertex.24681012foldsinterior verticesfacetscrease lengththe median facet falls from 2.6e-1 to 4.0e-3 of the sheet

upTo: 12, seeds: [3, 11, 19, 27, 41], view: "growth"

Folding adds and never subtractsInterior vertices, facets and total crease length against the number of random folds, averaged over five seeds. Each curve rises and none of them turns over: a fold can only add creases, so the pattern gets finer at every step and the facets between the creases get smaller. Every sheet in the sweep passes the flat-folding condition at every vertex.24681012foldsinterior verticesfacetscrease lengththe median facet falls from 2.6e-1 to 4.0e-3 of the sheet

upTo: 13, seeds: [7, 23, 31]

Folding adds and never subtractsInterior vertices, facets and total crease length against the number of random folds, averaged over five seeds. Each curve rises and none of them turns over: a fold can only add creases, so the pattern gets finer at every step and the facets between the creases get smaller. Every sheet in the sweep passes the flat-folding condition at every vertex.24681012foldsinterior verticesfacetscrease lengththe median facet falls from 2.1e-1 to 2.4e-3 of the sheet

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.

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.

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.

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.

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