Origami & Folding
what one sheet can do
Generator

The same vertex, drawn three ways

A generator in the curves and material library, called 13 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.

curved-vertex 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 vertex, drawn three waysOne vertex with four creases, drawn with the creases straight and then with them curved by two different amounts. The tangents at the vertex are identical in all of them and so are the sectors between them, which is where the conditions live.straight creasescurvature 1.4curvature 2.6the tangents are the same in every panelsectors 70.0°, 110.0°, 110.0°, 70.0°they sum to 360.0°, and alternately to 180.0° and 180.0°which is Kawasaki, on tangents rather than on lines

sub: "ruling-relation", r: 1

What folding can and cannot changeA circle drawn on a flat sheet and folded into a cone, at every fold angle from shallow to fully flat. The curvature measured in space rises without limit as the fold tightens, and the curvature within the surface does not move at all — because folding is an isometry, and an isometry cannot change what happens inside the paper. The difference between them is exactly what the fold angle buys.2030405060708000.511.522.5cone half-angle β, degrees — 90° is the unfolded sheetcurvature, in units of 1/rκ, in space1/(r sin β) — unboundedκ_g, in the surface1/r — flat, at every anglethe isometry, as a lineκ_n, out of the surfacecot β / r — all of the gainκ² = κ_g² + κ_n² to 3e-15true at every sample,not only at the ends

sub: "ruling-relation", r: 0.7, betas: [25, 40, 55, 70, 85]

What folding can and cannot changeA circle drawn on a flat sheet and folded into a cone, at every fold angle from shallow to fully flat. The curvature measured in space rises without limit as the fold tightens, and the curvature within the surface does not move at all — because folding is an isometry, and an isometry cannot change what happens inside the paper. The difference between them is exactly what the fold angle buys.3040506070800123cone half-angle β, degrees — 90° is the unfolded sheetcurvature, in units of 1/rκ, in space1/(r sin β) — unboundedκ_g, in the surface1/r — flat, at every anglethe isometry, as a lineκ_n, out of the surfacecot β / r — all of the gainκ² = κ_g² + κ_n² to 4e-15true at every sample,not only at the ends

sub: "ruling-relation", r: 0.5, betas: [30, 40, 50, 60, 70, 80, 90]

What folding can and cannot changeA circle drawn on a flat sheet and folded into a cone, at every fold angle from shallow to fully flat. The curvature measured in space rises without limit as the fold tightens, and the curvature within the surface does not move at all — because folding is an isometry, and an isometry cannot change what happens inside the paper. The difference between them is exactly what the fold angle buys.3040506070809001234cone half-angle β, degrees — 90° is the unfolded sheetcurvature, in units of 1/rκ, in space1/(r sin β) — unboundedκ_g, in the surface1/r — flat, at every anglethe isometry, as a lineκ_n, out of the surfacecot β / r — all of the gainκ² = κ_g² + κ_n² to 7e-15true at every sample,not only at the ends

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.

One curve and one number

Folding cannot change how curved a crease is within the surface — that is what an isometry means. Everything a curved fold produces is the curvature it adds out of the surface, and one number controls all of it.

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.

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