Origami & Folding
what one sheet can do
Generator

Two colours, and no choice about them

A generator in the flat-folding library, called 18 times across 6 essays. 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.

panel-colouring 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

Two colours, and no choice about themThe panels of a flat-foldable pattern, painted in the two colours the creases force on them. The colouring is computed by crossing creases and counting; the parity of the crease counts at the vertices is computed separately from the edges. They are the same fact twice.12 panels, two coloursno crease has the same colour on both sidesall 6 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge

pattern: "miura", cols: 4, rows: 3

Two colours, and no choice about themThe panels of a flat-foldable pattern, painted in the two colours the creases force on them. The colouring is computed by crossing creases and counting; the parity of the crease counts at the vertices is computed separately from the edges. They are the same fact twice.12 panels, two coloursno crease has the same colour on both sidesall 6 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge

pattern: "waterbomb", cols: 4, rows: 4, mask: 46

Two colours, and no choice about themThe panels of a flat-foldable pattern, painted in the two colours the creases force on them. The colouring is computed by crossing creases and counting; the parity of the crease counts at the vertices is computed separately from the edges. They are the same fact twice.52 panels, two coloursno crease has the same colour on both sidesall 25 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge

pattern: "preliminary", cols: 4, rows: 3

Two colours, and no choice about themThe panels of a flat-foldable pattern, painted in the two colours the creases force on them. The colouring is computed by crossing creases and counting; the parity of the crease counts at the vertices is computed separately from the edges. They are the same fact twice.8 panels, two coloursno crease has the same colour on both sidesall 1 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge

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.

A contradiction is even

A crease pattern's letters can demand a circle of panels each of which lies below the next, which is a proof that the sheet has no folded state. Every such circle found here — one thousand one hundred and forty-nine of them, across every family of patterns this collection draws — has an even number of panels in it, and none has four. Both facts are theorems rather than observations, and they come from opposite ends of the subject.

Bringing the other side to the front

Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.

Decided before the design

A colour change brings the reverse side of the paper to the front, and the usual account is that the two-colouring of the panels decides which panels are available. Measured on the site's own printed patterns, availability is not the constraint: both sides lie over more than ninety-nine per cent of most folded footprints. The other side is not scarce. It is under eight layers of paper.

The outline is mostly crease

The edge of a folded model is what a reader looks at, and almost none of it is the edge of the paper. Measured across five patterns, the sheet's own boundary accounts for between nothing and a third of the exposed edge; the rest is fold, and on a waterbomb tessellation the raw edge does not reach the outside at all.

The sheet has two sides

Read a crease pattern as a set of panels rather than a set of lines and a condition appears that no vertex theorem states: the panels take two colours, no crease has the same colour on both sides, and the colour is which face of the paper each panel ends up showing.

Which side arrives

A colour change is described as a choice: bring the reverse of the sheet to the front where the design wants it. On a pattern whose panels can be ordered, nobody chooses. The preliminary base shows the side that started face up over one part in a thousand of its own footprint, and two of its eight panels are the only ones in view at all.

Every generator · The flat-folding field · The patterns a reader can fold