Origami & Folding
what one sheet can do
Generator

A curved crease, cut into panels

A generator in the curves and material 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.

curved-rigid 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

A curved crease, cut into panelsOne curved crease and its straight-line approximations at 4, 8, 16 segments, with the sheet either side drawn as the flat panels each approximation demands. The panels grow narrower and more numerous and the picture improves; the total turning collected at the joints is identical in all of them, which is why no finite number of flat panels is a curved crease.the creaseno panels at all128.3° of turning4 segments8 panels128.3° of turning8 segments16 panels128.3° of turning16 segments32 panels128.3° of turningone crease, cut into flat pieceseach panel keeps its width across the crease and loses length along it as the count risesa finer approximation is a better picture and the same total kink, spread over more joints

view: "segments", segments: [4, 8, 16]

A curved crease, cut into panelsOne curved crease and its straight-line approximations at 4, 8, 16 segments, with the sheet either side drawn as the flat panels each approximation demands. The panels grow narrower and more numerous and the picture improves; the total turning collected at the joints is identical in all of them, which is why no finite number of flat panels is a curved crease.the creaseno panels at all128.3° of turning4 segments8 panels128.3° of turning8 segments16 panels128.3° of turning16 segments32 panels128.3° of turningone crease, cut into flat pieceseach panel keeps its width across the crease and loses length along it as the count risesa finer approximation is a better picture and the same total kink, spread over more joints

view: "segments", segments: [2, 4, 8]

A curved crease, cut into panelsOne curved crease and its straight-line approximations at 2, 4, 8 segments, with the sheet either side drawn as the flat panels each approximation demands. The panels grow narrower and more numerous and the picture improves; the total turning collected at the joints is identical in all of them, which is why no finite number of flat panels is a curved crease.the creaseno panels at all128.3° of turning2 segments4 panels128.3° of turning4 segments8 panels128.3° of turning8 segments16 panels128.3° of turningone crease, cut into flat pieceseach panel keeps its width across the crease and loses length along it as the count risesa finer approximation is a better picture and the same total kink, spread over more joints

view: "error", segments: [4, 8, 16]

Two ways of getting better, and one of notThree quantities against the number of straight segments a curved crease is approximated by, each as a fraction of its value at the coarsest approximation, on logarithmic axes. The distance from the true curve collapses, the turning at each joint falls in proportion to the joints there are, and the total turning does not move: it is a constant of the curve, and the panels can only divide it more finely.481632641280.0010.010.11straight segments the curved crease is cut intoas a fraction of its value at the coarsesttotal turningnever falls at allfitted slope 0.000turning per jointfalls as one over kfitted slope -1.000distance from the curvefalls as one over k squaredfitted slope -1.998every angle is measured off the chords built, and the two falling slopes are fitted rather than quotedthe flat series is the one that decides it: 128.3° of turning at 4 segments and at 128

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 curve has no panels

A rigid folding is a finite list of flat pieces joined along lines. A curved crease has no such list, and refining one does not help: the kink at each joint falls as one over the segment count, and the total of the kinks does not fall at all, because it is a constant of the curve.

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