Origami & Folding
what one sheet can do
Generator

Flat-foldability does not notice

A generator in the rigid folding library, called 6 times across 1 essay. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

error-field is one function. Everything below came out of it during this build, at arguments 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.

At its defaults

Flat-foldability does not noticeHow far the panels come apart against how far one row was moved. The mesh is exactly as flat-foldable at every point of this graph as it was at the start — the vertex conditions are satisfied to within rounding throughout — and the sheet stops folding rigidly the moment the displacement is anything but zero.6.6e-42.0e-34.7e-31.1e-2Kawasaki: 5e-14°, flatthe gaphow far the row was moved, as a fraction of its own stephow far two panels leave a shared cornerone quantity is a condition at a point and the other is a distance between panels, and only the second can see the mistake

dphi: 0.16, deltas: [0.02, 0.05, 0.09, 0.14], row: 2

Flat-foldability does not noticeHow far the panels come apart against how far one row was moved. The mesh is exactly as flat-foldable at every point of this graph as it was at the start — the vertex conditions are satisfied to within rounding throughout — and the sheet stops folding rigidly the moment the displacement is anything but zero.6.6e-42.0e-34.7e-31.1e-2Kawasaki: 5e-14°, flatthe gaphow far the row was moved, as a fraction of its own stephow far two panels leave a shared cornerone quantity is a condition at a point and the other is a distance between panels, and only the second can see the mistake

view: "rows", dphi: 0.12, deltas: [0.01, 0.04, 0.08, 0.12], row: 1, drive: 1

Flat-foldability does not noticeHow far the panels come apart against how far one row was moved. The mesh is exactly as flat-foldable at every point of this graph as it was at the start — the vertex conditions are satisfied to within rounding throughout — and the sheet stops folding rigidly the moment the displacement is anything but zero.1.0e-44.3e-49.5e-41.6e-3Kawasaki: 3e-14°, flatthe gaphow far the row was moved, as a fraction of its own stephow far two panels leave a shared cornerone quantity is a condition at a point and the other is a distance between panels, and only the second can see the mistake

dphi: 0.16, deltas: [0.02, 0.05, 0.09, 0.14], row: 2, view: "map"

The mistake is in a row; the disagreement is between columnsEach column's cosine divided by the first column's, read off every row. On the sheet that folds they are the same in every row, which is the rank-one condition written out. One row moved sideways, and every entry of that row changes — so the columns can no longer agree about a common ratio, and the failure has no single place.the ratio of each column's cosine to the first column'scol 1col 2col 3col 4col 5the sheet that folds1.0000-1.08221.0000-1.08221.00001.0000-1.08221.0000-1.08221.00001.0000-1.08221.0000-1.08221.0000row 1row 2row 3one row moved by 14 per cent1.0000-1.08221.0000-1.08221.00001.0000-1.13971.0000-1.13971.00001.0000-1.08221.0000-1.08221.0000row 1row 2row 3worst disagreement between rows: 4e-16 before, 0.0575 after

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

Where an error goes

A misplaced crease in a folded sheet has to be paid for somewhere, and this subject has two answers already — the error is folded too, and the hinge is where it ends up. There is a third. In a quadrilateral mesh a mistake in one row has no consequence in that row at all: it is felt by the columns, which is to say by every other row on the sheet.

Every generator · The rigid folding field · The patterns a reader can fold