Origami & Folding
what one sheet can do
Generator

One vertex, several sets of crease lengths

A generator in the flat-folding library, called 6 times across 1 essay. 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.

angles-not-lengths 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

One vertex, several sets of crease lengthsThe same two free sector angles drawn with the creases run out to wildly different lengths. Kawasaki, Maekawa and the big-little-big lemma read the angles and nothing else, so every one of these is the same vertex as far as any condition in the subject is concerned — and each folds into a different shape.sectors 80°, 55°, 100°, 125° in every one of them, and 4 assignments fold in every onelongest ÷ shortest 1.00footprint 0.806longest ÷ shortest 3.09footprint 0.911longest ÷ shortest 3.33footprint 0.623longest ÷ shortest 4.00footprint 1.782every one of them folds; their folded footprints differ by a factor of 2.86

view: "family"

One vertex, several sets of crease lengthsThe same two free sector angles drawn with the creases run out to wildly different lengths. Kawasaki, Maekawa and the big-little-big lemma read the angles and nothing else, so every one of these is the same vertex as far as any condition in the subject is concerned — and each folds into a different shape.sectors 80°, 55°, 100°, 125° in every one of them, and 4 assignments fold in every onelongest ÷ shortest 1.00footprint 0.806longest ÷ shortest 3.09footprint 0.911longest ÷ shortest 3.33footprint 0.623longest ÷ shortest 4.00footprint 1.782every one of them folds; their folded footprints differ by a factor of 2.86

view: "folded"

Two members of one family, foldedThe same four sector angles with two different sets of crease lengths, each folded flat by composing the reflections in its own creases. Both fold — the conditions cannot tell them apart — and the paper ends up covering different amounts of the table.longest crease ÷ shortest = 1.004 assignments foldfolded footprint 0.8054 layers at the deepest pointlongest crease ÷ shortest = 4.004 assignments foldfolded footprint 1.7824 layers at the deepest pointsame sectors, same conditions, same verdict — and the folded states are different objects

view: "stretch"

One crease stretched, and nothing that decides anything movesOne crease of a flat-foldable degree-four vertex is drawn longer and longer while the other three are held. The number of assignments that fold is the flat line; the area the folded vertex covers is the one that climbs. Both are plotted against their value at the shortest length, so the flat line is flat because nothing about foldability changed.0.40.60.811.21.41.61.8200.511.52length of the swept crease, in sheet unitsmeasured against the shortestfolded footprint · ×2.03assignments that fold · 4, throughout13 lengths, sectors held at 80°, 55°, 100°, 125° throughoutKawasaki's alternating sums never disagree by more than 1.8e-15 radians over the whole sweep

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.

The lengths are free

Kawasaki reads angles, Maekawa counts letters, and the big-little-big lemma compares one sector with its neighbours. Not one condition in the subject mentions how long a crease is — so a single vertex is not a pattern but a whole family of them, every member folding, no two folding into the same shape.

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