Origami & Folding
what one sheet can do
Generator

The same error, three directions

A generator in the rigid folding library, called 8 times across 2 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.

error-direction 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 same error, three directionsHow far a solved mesh is from closing after every one of its lengths is cut wrong by the same amount, against the size of that error on a sheet 150 millimetres across, in three directions. Across the surface of solutions the closure is lost in a fifth of a millimetre. Along it, the same error a hundred times larger costs less.00.20.40.60.81-7-6-5-4-3-2-1error in every length, millimetres on a 150 mm sheetclosure mismatch (powers of ten)across the surface of solutionsthe direction the closure's own derivative points ina direction chosen without regard to the surfacewhich has a component along bothalong the surface of solutionsone of the twelve directions the equations do not seethe same step costs 5,130 times as much one way as the other

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

The same error, three directionsHow far a solved mesh is from closing after every one of its lengths is cut wrong by the same amount, against the size of that error on a sheet 150 millimetres across, in three directions. Across the surface of solutions the closure is lost in a fifth of a millimetre. Along it, the same error a hundred times larger costs less.00.20.40.60.81-7-6-5-4-3-2-1error in every length, millimetres on a 150 mm sheetclosure mismatch (powers of ten)across the surface of solutionsthe direction the closure's own derivative points ina direction chosen without regard to the surfacewhich has a component along bothalong the surface of solutionsone of the twelve directions the equations do not seethe same step costs 5,130 times as much one way as the other

view: "space", seed: 11, rho: 0.6

How many directions there are of each kindThe lengths a quadrilateral mesh has, sorted by whether the closure conditions see a change in them. Four of the directions are constrained and the rest are not, so the surface of solutions is wide and an error is much more likely to point somewhere harmless than somewhere fatal.20 lengths · 4 constrained directions · 16 free ones4 directions the closure sees — a step here is paid for in proportion16 it does not — a step here changes nothing to first orderwhich is why one figure of a fifth of a millimetre describes neither

view: "space", seed: 23, rho: 0.6

How many directions there are of each kindThe lengths a quadrilateral mesh has, sorted by whether the closure conditions see a change in them. Four of the directions are constrained and the rest are not, so the surface of solutions is wide and an error is much more likely to point somewhere harmless than somewhere fatal.20 lengths · 4 constrained directions · 16 free ones4 directions the closure sees — a step here is paid for in proportion16 it does not — a step here changes nothing to first orderwhich is why one figure of a fifth of a millimetre describes neither

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.

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