Origami & Folding
what one sheet can do
Generator

The allowance a solved mesh has, at each point of its fold

A generator in the rigid folding library, called 6 times across 1 essay. 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.

allowance-motion 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

The allowance a solved mesh has, at each point of its foldHow far a quadrilateral mesh may be cut wrong before its closure fails, measured all the way along the motion rather than at the one fold angle a tolerance is normally quoted at. It falls by an order of magnitude between the flat sheet and the packed one.the height is the allowance in millimetres on a 150 mm sheetthe horizontal axis is the fold angle of the driven crease, in radians0.425 mm0.033 mm0.32.4fold angle of the driven creasea budget of 0.02 radians on a 150 mm sheet12.9 times less allowance at the closed endthe mesh is most forgiving where it is doing least

view: "allowance", seed: 11, budget: 0.02

The allowance a solved mesh has, at each point of its foldHow far a quadrilateral mesh may be cut wrong before its closure fails, measured all the way along the motion rather than at the one fold angle a tolerance is normally quoted at. It falls by an order of magnitude between the flat sheet and the packed one.the height is the allowance in millimetres on a 150 mm sheetthe horizontal axis is the fold angle of the driven crease, in radians0.425 mm0.033 mm0.32.4fold angle of the driven creasea budget of 0.02 radians on a 150 mm sheet12.9 times less allowance at the closed endthe mesh is most forgiving where it is doing least

view: "surface", seed: 11

The surface of solutions does not move as the sheet foldsA direction found at one fold angle, measured against the closure equations at another. The stiff direction stays entirely inside the equations' row space and the free one stays entirely outside it, at every pair of angles tried.4 equations in 20 lengths, so the solutions are a surface 16 directions widehow much of a direction found at one angle lies inside the row space at anotherfrom → tostiff direction · free direction0.3 → 0.51.000000000 · 5.5e-80.3 → 0.81.000000000 · 4.5e-80.3 → 1.11.000000000 · 5.5e-80.3 → 1.51.000000000 · 4.0e-80.3 → 2.01.000000000 · 5.5e-80.3 → 2.51.000000000 · 5.0e-80.5 → 0.31.000000000 · 5.5e-80.5 → 0.81.000000000 · 1.2e-8one and zero, to twelve figures, in every direction of travel

view: "multiplier", seed: 11, budget: 0.02

The sensitivity, against the fold-angle multiplierThe closure's sensitivity to a cutting error, divided by the tangent of half the fold angle. Over the range a builder uses it is constant to within two per cent, so a single number and a known function of the fold angle cover the whole motion.the bar is the sensitivity divided by tan(ρ/2)it varies by 24 per cent over the whole sweep and by far less over the buildable partρ = 0.300.908 ÷ tan(ρ/2) = 6.007ρ = 0.451.373 ÷ tan(ρ/2) = 6.001ρ = 0.601.853 ÷ tan(ρ/2) = 5.991ρ = 0.802.525 ÷ tan(ρ/2) = 5.972ρ = 1.003.248 ÷ tan(ρ/2) = 5.946ρ = 1.204.043 ÷ tan(ρ/2) = 5.909ρ = 1.505.425 ÷ tan(ρ/2) = 5.824ρ = 1.807.154 ÷ tan(ρ/2) = 5.677ρ = 2.109.423 ÷ tan(ρ/2) = 5.405ρ = 2.4012.462 ÷ tan(ρ/2) = 4.845

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.

The allowance is spent at the end

A tolerance on a solved mesh was priced at one fold angle, because that is where a tolerance is priced. The surface of solutions turns out not to move as the sheet folds — the free directions at a third of a radian are the free directions at two and a half, to twelve figures — and the price of leaving it rises by a factor of thirteen along the way.

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