Origami & Folding
what one sheet can do
Generator

The same curvature, kept in one place or spent everywhere

A generator in the folding nobody designed library, called 7 times across 3 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.

deficit-two-ways 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 curvature, kept in one place or spent everywhereRemoving a wedge and closing the gap makes a cone: all of its curvature sits at the apex and the rest of the surface is flat, so a walk that avoids one point measures nothing. A growth field carrying the same total has no special point at all. Folding concentrates; growth spreads.a crease — concentrated at a pointdeficit 0.5236 rad, all of it hereflat everywhere elsegrowth — spread over the areatotal 0.5236 rad, none of it anywherecurved at every pointa 30° wedge removed, against Ω = 1 − 0.0400 r² · same total curvature, 0.5236

wedgeDeg: 30

The same curvature, kept in one place or spent everywhereRemoving a wedge and closing the gap makes a cone: all of its curvature sits at the apex and the rest of the surface is flat, so a walk that avoids one point measures nothing. A growth field carrying the same total has no special point at all. Folding concentrates; growth spreads.a crease — concentrated at a pointdeficit 0.5236 rad, all of it hereflat everywhere elsegrowth — spread over the areatotal 0.5236 rad, none of it anywherecurved at every pointa 30° wedge removed, against Ω = 1 − 0.0400 r² · same total curvature, 0.5236

wedgeDeg: 45

The same curvature, kept in one place or spent everywhereRemoving a wedge and closing the gap makes a cone: all of its curvature sits at the apex and the rest of the surface is flat, so a walk that avoids one point measures nothing. A growth field carrying the same total has no special point at all. Folding concentrates; growth spreads.a crease — concentrated at a pointdeficit 0.7854 rad, all of it hereflat everywhere elsegrowth — spread over the areatotal 0.7854 rad, none of it anywherecurved at every pointa 45° wedge removed, against Ω = 1 − 0.0588 r² · same total curvature, 0.7854

wedgeDeg: 90

The same curvature, kept in one place or spent everywhereRemoving a wedge and closing the gap makes a cone: all of its curvature sits at the apex and the rest of the surface is flat, so a walk that avoids one point measures nothing. A growth field carrying the same total has no special point at all. Folding concentrates; growth spreads.a crease — concentrated at a pointdeficit 1.5708 rad, all of it hereflat everywhere elsegrowth — spread over the areatotal 1.5708 rad, none of it anywherecurved at every pointa 90° wedge removed, against Ω = 1 − 0.1111 r² · same total curvature, 1.5708

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 crease carries no curvature

A fold looks like the sharpest curvature a sheet could have, and intrinsically it has none at all. Developability — the first of the four conditions this site's checker runs — is exactly the statement that folding an uncut sheet creates no curvature anywhere, including at the creases.

A straight tuck is a cone point

A tuck with straight edges hides length in proportion to how far past its start it has gone, which is a cone's law and not a sphere's. Started at the centre, straight tucks make a cone. Started at several radii, they hide length in a broken line that follows a sphere's cubic, and the worst shortfall falls as the square of the number of starting radii: 36.9 per cent of the rim's hiding from one start, 12.3 from two, 3.3 from four, 0.8 from eight. Every start is three creases at a point, which is a vertex that cannot fold flat — and it is exactly where the gathered sheet's curvature goes.

The test measures the rim

Flattening the specimen is the right test and the measurement anybody actually makes on a flattened specimen is a boundary one — how far the margin overruns its chord. Total curvature is a boundary quantity too: it equals minus two pi R times the growth profile's slope at the rim, and nothing else about the interior survives into it. So a sheet can be curved everywhere and integrate to nothing, and the test reports it flat.

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