Origami & Folding
what one sheet can do
Generator

What each mesh's reason for folding survives

A generator in the rigid folding 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.

mesh-tolerance 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

What each mesh's reason for folding survivesEvery length of each mesh made wrong by the stated amount, and what is left of the closure. The solved mesh folds because an equation holds; the Miura folds because one crease family runs straight through every vertex, which a badly cut sheet still does.the solved mesh, cut wrong byworst mismatch left, radians0.017 mm0.00210.051 mm0.00580.169 mm0.01990.508 mmno closure at alla Miura, cut wrong by0 per cent2e-141 per cent6e-155 per cent4e-15

view: "joint", seed: 11, rho: 0.6, budget: 0.02

What each mesh's reason for folding survivesEvery length of each mesh made wrong by the stated amount, and what is left of the closure. The solved mesh folds because an equation holds; the Miura folds because one crease family runs straight through every vertex, which a badly cut sheet still does.the solved mesh, cut wrong byworst mismatch left, radians0.017 mm0.00210.051 mm0.00580.169 mm0.01990.508 mmno closure at alla Miura, cut wrong by0 per cent2e-141 per cent6e-155 per cent4e-15

view: "joint", seed: 11, rho: 0.6, mm: 150

What each mesh's reason for folding survivesEvery length of each mesh made wrong by the stated amount, and what is left of the closure. The solved mesh folds because an equation holds; the Miura folds because one crease family runs straight through every vertex, which a badly cut sheet still does.the solved mesh, cut wrong byworst mismatch left, radians0.017 mm0.00210.051 mm0.00580.169 mm0.01990.508 mmno closure at alla Miura, cut wrong by0 per cent2e-141 per cent6e-155 per cent4e-15

view: "allowance", seed: 11, rho: 0.6, mm: 150, budget: 0.02

The tolerance, in millimetresHow far the closure moves as the sheet is cut less and less accurately. The horizontal line is a mismatch of two hundredths of a radian, which is about a degree; where the curve crosses it is how accurately the sheet has to be cut.00.050.10.150.20.250.30.350.40.450.500.0050.010.0150.020.025error in every length, millimetres on a 150 mm sheetworst mismatch, radiansa degree of mismatch

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 tolerance is a direction

Cut a solved mesh a fifth of a millimetre wrong and its closure is gone. That is true of the errors it was tried with and false of errors in general: the solutions form a surface sixteen directions wide, an error along it costs five thousand times less than the same error across it, and the fifth of a millimetre is the allowance in one direction out of twenty.

Solved is not built

A mesh that folds because an equation holds and a mesh that folds because one crease family runs straight through every vertex are not two examples of the same thing. Cut a Miura's every dimension five per cent wrong and it still folds exactly. Cut a solved general mesh a fifth of a millimetre wrong on a 150 mm sheet and the closure is gone.

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