Origami & Folding
what one sheet can do
Generator

Every interior vertex in the library, by degree

A generator in the flat-folding library, called 7 times across 2 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

degree-census is one function. Everything below was drawn by it when the site was last published, at settings 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.

With nothing specified

Every interior vertex in the library, by degreeHow many creases meet at each interior vertex of every pattern in the library, counted together. The odd columns are empty and the column at two is not drawn at all: a vertex where creases stopped rather than passed through would be one of those, and the paper cannot fold there.interior vertices, by number of creases meeting therenone3odd594evennone5odd326evennone7odd18even8 patterns, 92 interior vertices, and not one of them with an odd number of creases

view: "library"

Every interior vertex in the library, by degreeHow many creases meet at each interior vertex of every pattern in the library, counted together. The odd columns are empty and the column at two is not drawn at all: a vertex where creases stopped rather than passed through would be one of those, and the paper cannot fold there.interior vertices, by number of creases meeting therenone3odd594evennone5odd326evennone7odd18even8 patterns, 92 interior vertices, and not one of them with an odd number of creases

view: "odd", degree: 3

Every marking of a three-crease vertex, and none of them foldsAll the mountain-and-valley assignments of one degree-three interior vertex, with the two counts Maekawa compares. Their difference is odd whatever the letters are, so it is never two, and the search that would find a foldable one returns nothing.all 8 assignments of a 3-crease interior vertexmountains and valleys must differ by exactly two, and here they differ by an odd numberVVV0M 3VMaekawa: differ by 3Kawasaki: odd number of creases (3)MVV1M 2VMaekawa: differ by 1Kawasaki: odd number of creases (3)VMV1M 2VMaekawa: differ by 1Kawasaki: odd number of creases (3)MMV2M 1VMaekawa: differ by 1Kawasaki: odd number of creases (3)VVM1M 2VMaekawa: differ by 1Kawasaki: odd number of creases (3)MVM2M 1VMaekawa: differ by 1Kawasaki: odd number of creases (3)VMM2M 1VMaekawa: differ by 1Kawasaki: odd number of creases (3)MMM3M 0VMaekawa: differ by 3Kawasaki: odd number of creases (3)Kawasaki does not even get a hearing: it needs an even number of sectors to alternatethe count that has to differ by two is a sum of the same parity as the degree

view: "minimum", sectors: [140, 110, 110], degree: 3

Three creases cannot closeA degree-three interior vertex, with the side of the paper each sector shows on the left and the folded cross-section on the right. Crossing a crease turns the sheet over, so the shading has to alternate all the way round; with an odd number of creases two neighbours end up alike, and the folded walk comes back pointing the wrong way.the sides the sectors show140°110°110°two sectors share a side across the marked creasethe folded cross-sectionthe walk ends 140° from where it startedno set of 3 angles summing to 360° does better — it is the parity that fails, not the angles

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

A cut is not local

Cutting one crease of a square twist takes its letterings from sixteen mutually unreachable pieces to two. The cut crease is one of the four that were settled when the pattern was drawn — and it takes two others with it, because the vertices it releases were the far ends of those. Even a cut along a crease that was never settled quarters the count.

Nothing meets at three

Every interior vertex of a flat-foldable pattern carries an even number of creases and at least four. So a crease cannot stop in the middle of the sheet, three creases cannot meet anywhere, and every crease pattern in the subject ends up looking the same way — all crossings and no stars.

All the figures · The flat-folding field · The patterns a reader can fold