Origami & Folding
what one sheet can do
Generator

Conditions arrive with the interior

A generator in the flat-folding library, called 15 times across 3 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.

patch-conditions 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

Conditions arrive with the interiorInterior vertices per crease as a patch of one tessellation grows. Every interior vertex is four conditions and every boundary vertex is none, so a small patch is not a small version of the pattern — it is a much less constrained one.12345600.10.20.30.4patch, in cells acrossinterior vertices per creasewaterbomb1 interior vertices at 1 across, 61 at 6the ratio has a ceiling and approaches it from below, so a big patch is the honest test

view: "rules", ruleSizes: [2,2, 3,3, 4,4, 5,5]

Rules that only fold while nobody is lookingEvery one of the 512 repeating mountain-and-valley rules for the waterbomb tessellation, put past all four conditions on patches of four sizes. The ones that drop out between the smallest patch and the next were never foldable; the small patch simply had no interior vertex for them to fail at.562 × 24 cells323 × 39 cells324 × 416 cells325 × 525 cellsrepeating rules that pass every condition, out of 51224 rules pass on the smallest patch and on none of the others

view: "count", pattern: "waterbomb", sizes: [1, 2, 3, 4, 5, 6], mask: 46

Conditions arrive with the interiorInterior vertices per crease as a patch of one tessellation grows. Every interior vertex is four conditions and every boundary vertex is none, so a small patch is not a small version of the pattern — it is a much less constrained one.12345600.10.20.30.4patch, in cells acrossinterior vertices per creasewaterbomb1 interior vertices at 1 across, 61 at 6the ratio has a ceiling and approaches it from below, so a big patch is the honest test

view: "boundary", pattern: "waterbomb", cols: 3, rows: 3

Which vertices of the waterbomb carry a conditionEvery vertex of one pattern, marked by whether the local conditions apply to it. A vertex strictly inside the sheet has a full turn of paper and carries developability, Kawasaki, Maekawa and the big-little-big lemma; a vertex on the edge has less than a turn and carries none of them. The ones drawn in colour are those a test that looks for an attached boundary edge, rather than for the edge itself, gets wrong.waterbomb: 13 interior, 12 on the edgethe patternwaterbombvertices with three creases or more25interior, carrying four conditions each13on the edge, carrying none12inside a boundary edge, not at its end0read as interior by the old test0no vertex here sits inside a boundary edge, which is why this shape never met the bugfilled: a full turn of paper, and every condition. open: less, and none

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 unit that folds is not a tessellation

Of the 512 repeating rules for the waterbomb tessellation, 56 pass every condition on a two-by-two patch and 32 pass on every larger one. The twenty-four that die were never foldable — the small patch simply contained one of the four kinds of vertex the pattern makes, and the failures were at the other three.

Thirty-two rules, one object

Five hundred and twelve repeating rules for the waterbomb tessellation, fifty-six that pass on a small patch, thirty-two that pass on one containing every kind of vertex. Fold all thirty-two and compare their panels: the same panels, in the same places, with the same areas, every time. The rules are thirty-two labels on one object, and a count of them has counted the labels.

Where the paper stops

Every flat-folding theorem is a statement about a full turn of paper, so a vertex at the edge of the sheet is subject to none of them. Cutting a patch out of a pattern removes conditions rather than preserving them, and a small enough patch has almost none left.

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