Origami & Folding
what one sheet can do
Generator

How much coincidence a flat-foldable pattern is

A generator in the flat-folding library, called 14 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.

kawasaki-residual 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

How much coincidence a flat-foldable pattern isSectors drawn at random at one vertex, two vertices and three, and the fraction that come within a given tolerance of Kawasaki's condition. Each vertex is one equation, so each vertex costs another factor of the tolerance, and the lines are the powers.-3-2.5-2-1.5-1-8-6-4-20tolerance (log₁₀ radians)fraction inside it (log₁₀)1 vertex · slope 1.002 vertices · slope 2.013 vertices · slope 3.0140,000 random vertices, none of them constructed to fold and none of them folding

view: "distance", trials: 40000

How far from folding a random vertex isThe share of random four-crease vertices that would satisfy Kawasaki if each of their four sectors moved by no more than the amount on the axis. Failing is not a yes or a no — the alternating sum is a distance, and a quarter of it moves every sector at once onto a vertex that folds — and the distances are large: the median random vertex is over thirty degrees per sector away, and fewer than one in fifty is within a degree.0.5°1%1°2%2°3%5°8%10°16%20°33%how far each sector would have to moveshare that would fold40,000 random four-crease verticesthe median vertex is 31.31° per sector from folding, the mean 33.91°none of them folds, and almost none of them nearly does either

view: "random", trials: 40000

How much coincidence a flat-foldable pattern isSectors drawn at random at one vertex, two vertices and three, and the fraction that come within a given tolerance of Kawasaki's condition. Each vertex is one equation, so each vertex costs another factor of the tolerance, and the lines are the powers.-3-2.5-2-1.5-1-8-6-4-20tolerance (log₁₀ radians)fraction inside it (log₁₀)1 vertex · slope 1.002 vertices · slope 2.013 vertices · slope 3.0140,000 random vertices, none of them constructed to fold and none of them folding

view: "jitter", jitters: [0.002, 0.005, 0.01, 0.02, 0.04], cols: 4, rows: 3

A pattern stops folding as soon as it is nudgedThe Miura fold with every vertex moved by a small random amount, and the average by which Kawasaki's condition then fails. It grows in proportion to the disturbance and it is never zero: not one of the vertices survives a nudge of two thousandths of a cell.00.010.020.030.0400.010.020.030.040.05vertices moved by (cell widths)Kawasaki fails by (radians)6 interior vertices, and at every disturbance 0 of them still foldthe undisturbed pattern sits at exactly zero, where nothing lands by accident

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.

A near miss is nearly as rare

Flat-foldability is a coincidence of measure zero, which is usually where the argument stops. Measure how far a random vertex is from folding rather than whether it does, and the answer is thirty-one degrees a sector — so the tolerance real paper has does not buy back anything at all, and a pattern that nearly folds had to start near one that did.

A no costs more than a yes

When a folding question comes back yes, it comes back with an object: a labelling, a stacking, a folded state that anybody can check in one pass. When it comes back no, it comes back with nothing but the assurance that a search looked everywhere — and that assurance is the first thing to break.

Almost every pattern fails

Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured.

All the figures · The flat-folding field · The patterns a reader can fold