Origami & Folding
what one sheet can do
Generator

Two ways to be a degree out

A generator in the rigid folding library, called 10 times across 4 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-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

Two ways to be a degree outHow far the far end of a folded strip lands from where it belongs, against the number of creases, for the same size of error applied consistently and applied at random. The first line is straight and the second is a square root, and the gap between them is the whole difference between a machine out of calibration and a machine that is merely imprecise.05101520253000.050.10.150.20.250.3creasesdrift, in panel widthsthe same way each timeat randomeach crease 0.5° out · 40 strips averaged for the random casethe drift is a composition of reflections, and would be the same on paper

view: "laws", counts: [2, 4, 8, 16, 24, 32], epsDeg: 0.5, samples: 40

Two ways to be a degree outHow far the far end of a folded strip lands from where it belongs, against the number of creases, for the same size of error applied consistently and applied at random. The first line is straight and the second is a square root, and the gap between them is the whole difference between a machine out of calibration and a machine that is merely imprecise.05101520253000.050.10.150.20.250.3creasesdrift, in panel widthsthe same way each timeat randomeach crease 0.5° out · 40 strips averaged for the random casethe drift is a composition of reflections, and would be the same on paper

view: "laws", counts: [2, 4, 8, 16, 32, 48], epsDeg: 1, samples: 40

Two ways to be a degree outHow far the far end of a folded strip lands from where it belongs, against the number of creases, for the same size of error applied consistently and applied at random. The first line is straight and the second is a square root, and the gap between them is the whole difference between a machine out of calibration and a machine that is merely imprecise.01020304000.20.40.60.8creasesdrift, in panel widthsthe same way each timeat randomeach crease 1° out · 40 strips averaged for the random casethe drift is a composition of reflections, and would be the same on paper

view: "stack", n: 16, epsDeg: 1, counts: [2, 4, 8, 16, 24, 32], samples: 40

The same stack, folded three waysA strip of sixteen creases folded flat, with the panels drawn where the fold puts them. The first has every crease square. The second has each one out by the same small angle and the stack fans; the third has the same size of error scattered and the stack merely blurs.square creasesevery crease 1° the same way1° at randomevery one of the three is a valid crease pattern; two of them are not the pattern that was wanted

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.

Error is folded too

A folded position is a composition of reflections, and a reflection in a line that is slightly off turns everything beyond it by twice as much. So an error does not stay where it was made — and whether it grows with the crease count or with its square root depends on whether it is the same error every time.

Exact is not accurate

There are two ways of dividing a strip into equal parts: a ladder that lands on the fraction as a rational number, and Fujimoto's method, which never arrives. Read as mathematics that settles it. Read as instructions for somebody with a sheet of paper it settles nothing, and past four parts the method that never arrives is the one whose crease lands nearer the mark.

How deep is a crossing

A crossing is a verdict with no middle: two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched. Measured on the patches where they occur, the shallowest crossing runs 0.16 mm past the end of the crease it meets, on a sheet 150 mm across. Five of the forty-seven are under half a millimetre, which is thinner than the line a pencil draws.

Nowhere to put the error

Paper takes a misplaced crease and spreads it along its whole length as a curvature nobody notices. A panel is flat by definition and cannot, so the error arrives at the hinge — and the room to receive it is a length that has to be drawn, is paid for in fold angle, and has to grow with the crease count.

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