Origami & Folding
what one sheet can do
Generator

Where the conditions stop being the answer

A generator in the flat-folding library, called 14 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.

crimp-gap 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

Where the conditions stop being the answerFor each row, how many mountain-and-valley labellings satisfy all four local conditions, how many of those have a flat folded state, and the difference. At degree four the two columns are the same number; above it they are not.pass all fouractually foldthe gapdegree 4373 vertices14921492nonedegree 6329 vertices32082632576degree 858 vertices1440928512the gap is what a checker built from the four conditions would certify and a folder could not make

view: "vertex", slug: "rosette"

the preliminary base's centre: what passes, and what foldsEvery mountain-and-valley labelling of a vertex with sectors of 45°, 45°, 45°, 45°, 45°, 45°, 45°, 45°, sorted twice: by whether it satisfies developability, Kawasaki, Maekawa and the big-little-big lemma, and by whether a stacking of its sectors exists at all.the preliminary base's centredegree eight, and the vertex at the middle of the first base anybody foldsevery assignment256passes all four conditions112has a flat folded state112the conditions and the answer agree at every labelling

view: "vertex", slug: "waterbomb"

the waterbomb tessellation's odd vertex: what passes, and what foldsEvery mountain-and-valley labelling of a vertex with sectors of 90°, 45°, 45°, 90°, 45°, 45°, sorted twice: by whether it satisfies developability, Kawasaki, Maekawa and the big-little-big lemma, and by whether a stacking of its sectors exists at all.the waterbomb tessellation's odd vertexdegree six, and this site prints nine of them on one sheetevery assignment64passes all four conditions30has a flat folded state1812 labellings satisfy every condition the subject has and have no flat folded state

view: "degrees", degrees: [4, 6, 8]

Where the conditions stop being the answerFor each row, how many mountain-and-valley labellings satisfy all four local conditions, how many of those have a flat folded state, and the difference. At degree four the two columns are the same number; above it they are not.pass all fouractually foldthe gapdegree 4373 vertices14921492nonedegree 6329 vertices32082632576degree 858 vertices1440928512the gap is what a checker built from the four conditions would certify and a folder could not make

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.

Crimp it away and ask again

Four conditions decide whether a vertex folds flat, and they decide it exactly at a vertex whose sectors are all different sizes. Everywhere else they over-count: two markings of every tied four-crease vertex, twelve of the degree-six vertex this site prints nine of on one sheet. What decides the case is not a fifth condition but a procedure — fold the smallest sector away and ask the smaller vertex.

The whole alphabet of a grid

Box pleating is defended as a trade — give up packing efficiency, buy creases that land where they should. There is a third thing it buys and it is much stronger than either: on a forty-five degree grid there are exactly six kinds of interior vertex a flat-foldable design can contain, ever. On a thirty degree grid there are thirty.

Which vertices are the random ones

Every measurement on this site that begins 'over 373 random degree-four vertices' is a statement about a population nobody declared. There is no canonical way to pick a crease pattern at random, four defensible ways of doing it disagree about the same three questions by factors rather than by margins, and the disagreement reaches a sentence this site has published as though it were general.

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