Origami & Folding
what one sheet can do
Generator

A curved crease, and the rulings it forces

A generator in the curves and material library, called 9 times across 5 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.

curved-developable 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, and the rulings it forcesConcentric arcs on a flat sheet, and the surface they produce. Each band becomes a cone, the bands alternate which way they open, and the map from the flat sheet to the surface preserves every distance exactly — which is checked here rather than assumed. The straight lines on the surface are the cones' generators, and on the flat sheet they are radii.the patternwhat the sheet doesconcentric arcs with their rulings drawn as radii; the metric matches the flat sheet to 5e-7so nothing here is stretching — every point of the surface is where folding alone can put itthe flat-folding theorems say nothing about any of this: they are about straight creases meeting at a point

bands: 4, beta: 62, span: 150

A curved crease, and the rulings it forcesConcentric arcs on a flat sheet, and the surface they produce. Each band becomes a cone, the bands alternate which way they open, and the map from the flat sheet to the surface preserves every distance exactly — which is checked here rather than assumed. The straight lines on the surface are the cones' generators, and on the flat sheet they are radii.the patternwhat the sheet doesconcentric arcs with their rulings drawn as radii; the metric matches the flat sheet to 5e-7so nothing here is stretching — every point of the surface is where folding alone can put itthe flat-folding theorems say nothing about any of this: they are about straight creases meeting at a point

bands: 4, beta: 62, span: 150

A curved crease, and the rulings it forcesConcentric arcs on a flat sheet, and the surface they produce. Each band becomes a cone, the bands alternate which way they open, and the map from the flat sheet to the surface preserves every distance exactly — which is checked here rather than assumed. The straight lines on the surface are the cones' generators, and on the flat sheet they are radii.the patternwhat the sheet doesconcentric arcs with their rulings drawn as radii; the metric matches the flat sheet to 5e-7so nothing here is stretching — every point of the surface is where folding alone can put itthe flat-folding theorems say nothing about any of this: they are about straight creases meeting at a point

bands: 6, beta: 55

A curved crease, and the rulings it forcesConcentric arcs on a flat sheet, and the surface they produce. Each band becomes a cone, the bands alternate which way they open, and the map from the flat sheet to the surface preserves every distance exactly — which is checked here rather than assumed. The straight lines on the surface are the cones' generators, and on the flat sheet they are radii.the patternwhat the sheet doesconcentric arcs with their rulings drawn as radii; the metric matches the flat sheet to 5e-7so nothing here is stretching — every point of the surface is where folding alone can put itthe flat-folding theorems say nothing about any of this: they are about straight creases meeting at a point

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.

The sculptors got there first

Curved-crease folding produced its best objects decades before anybody could compute one. The surfaces were made by hand, the ruling lines that determine them were not calculated until much later, and the mathematics has been catching up ever since.

What a flat sheet can become

A sheet that cannot stretch cannot become a sphere. That much belongs to differential geometry; what belongs to folding is the three ways round it — seams, curved creases, and a few percent of stretch — and what each one costs.

Where curved creases meet

A curved-crease design looks like a smooth object and its constraints are not smooth. They live at the finitely many points where creases cross, and at each of those the conditions are about the creases' tangent directions — the curvature does not appear in them at all.

Where the rulings run out

A curved fold's surface is made of straight lines leaving the crease, and the lines are not parallel, so they cross. Past the first crossing there is no surface: two points of the paper have been sent to one point of space. The boundary is a curve nobody drew, no crease pattern shows it, and it sits at the sine of the ruling angle times the crease's own tightest radius — on every curve tried.

All the figures · The curves and material field · The patterns a reader can fold