Origami & Folding
what one sheet can do
Generator

What survives the local conditions

A generator in the flat-folding library, called 13 times across 4 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.

assignment-census 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

What survives the local conditionsFor each pattern: how many mountain-and-valley assignments there are, and how many of them satisfy every condition at every vertex. The filter is severe and it is not a decision — what passes is still an exponentially large set, and every member of it still has to be checked globally.degree-4 vertex4 of 1625.0% · 4 creasespreliminary base112 of 25643.8% · 8 creasesmiura 2×28 of 1650.0% · 4 creasesmiura 3×232 of 12825.0% · 7 creasesmiura 3×3256 of 4,0966.3% · 12 creasesevery count enumerated, none estimatedthe share falls as the pattern grows, and the count still rises

cases: [degree-4 vertex, miura 3×2, miura 3×3, miura 4×3]

What survives the local conditionsFor each pattern: how many mountain-and-valley assignments there are, and how many of them satisfy every condition at every vertex. The filter is severe and it is not a decision — what passes is still an exponentially large set, and every member of it still has to be checked globally.degree-4 vertex4 of 1625.0% · 4 creasesmiura 3×232 of 12825.0% · 7 creasesmiura 3×3256 of 4,0966.3% · 12 creasesmiura 4×32,048 of 131,0721.6% · 17 creasesevery count enumerated, none estimatedthe share falls as the pattern grows, and the count still rises

view: "entailed"

One letter settles almost nothingFor each pattern: how many creases it has in its interior, and how many of them follow from choosing one. The pale part of each bar is what the local conditions entail; the rest is what they leave open. On every pattern here the entailed part is a few creases out of dozens or hundreds.what the vertex conditions settle once one crease is chosenpatternsettled, against what is therepreliminary base1 of 81 vertices still choosingmiura 6×41 of 3815 vertices still choosingwaterbomb 4×41 of 7625 vertices still choosingyoshimura 6×51 of 8422 vertices still choosingsquare twist grid3 of 14464 vertices still choosingtriangular twist grid2 of 236104 vertices still choosingKawasaki was settled by the angles before a letter was written; the letters are what is left, and they are nearly all left

cases: [degree-4 vertex, preliminary base, miura 2×2, miura 3×2, miura 3×3]

What survives the local conditionsFor each pattern: how many mountain-and-valley assignments there are, and how many of them satisfy every condition at every vertex. The filter is severe and it is not a decision — what passes is still an exponentially large set, and every member of it still has to be checked globally.degree-4 vertex4 of 1625.0% · 4 creasespreliminary base112 of 25643.8% · 8 creasesmiura 2×28 of 1650.0% · 4 creasesmiura 3×232 of 12825.0% · 7 creasesmiura 3×3256 of 4,0966.3% · 12 creasesevery count enumerated, none estimatedthe share falls as the pattern grows, and the count still rises

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.

Hardness is about the worst one

Flat-foldability is NP-hard, and every crease pattern on this site is decided in under a second. Both are true, and holding them together is the difference between using the result and repeating it: hardness is a statement about the worst instance a family contains, and nobody folds the worst one.

How little the conditions decide

Local is not global is a statement about sufficiency: every vertex can pass and the sheet still fail. There is a sharper complaint available, and it is about strength. Fix one crease of a tessellation and propagate every condition the subject has to a fixed point: three creases out of a hundred and fifty-eight follow, and sixty-six vertices are still holding more than one answer.

How many assignments fold

The local conditions throw away most of the ways a pattern could be creased. They throw away a smaller and smaller fraction as the pattern grows, and what survives grows faster than what is discarded — which is why a strong filter is not a decision procedure.

The lettering that folds nowhere

The conditions at a vertex admit 256 letterings of the square twist. Eight of them have a folded state. The other 248 satisfy developability, Kawasaki, Maekawa and the big-little-big lemma at every vertex of the pattern and cannot be folded by anyone — and this site printed one of them for years, at true scale, with instructions to fold it first.

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