Origami & Folding
what one sheet can do
Generator

Same arrangement, same letterings

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.

order-type 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

Same arrangement, same letteringsVertices drawn at random and sorted by which of their sectors are strictly smallest. Inside each group every vertex admits exactly the same letterings — the same list, not merely the same number of them — and a count that never sees a vertex reproduces it.smallest sectors atverticesletteringspredictedfoldpositions 1, 443888positions 2, 525888positions 0, 325888positions 0, 417888positions 1, 314888positions 2, 413888positions 3, 512888positions 1, 511888the fourth column is computed by walking the cycle with no vertex present

view: "minima", angles: [40, 95, 25, 110, 60, 30]

All the conditions look atOne vertex, with the sectors that are strictly smaller than both their neighbours shaded. Kawasaki and developability are settled by the angles before any letter is written, Maekawa mentions no angle at all, and the only thing left reads this shading and nothing else.40°95°25°110°60°30°2 strictly smallest sectors, at 25° and 30°every vertex with the same shading admits exactly the same letterings

view: "groups", degree: 6, trials: 260

Same arrangement, same letteringsVertices drawn at random and sorted by which of their sectors are strictly smallest. Inside each group every vertex admits exactly the same letterings — the same list, not merely the same number of them — and a count that never sees a vertex reproduces it.smallest sectors atverticesletteringspredictedfoldpositions 1, 443888positions 2, 525888positions 0, 325888positions 0, 417888positions 1, 314888positions 2, 413888positions 3, 512888positions 1, 511888the fourth column is computed by walking the cycle with no vertex present

view: "count"

The count, with no vertex in itA degree-four vertex, with the letterings that satisfy Maekawa and the lemma counted from the arrangement of its smallest sectors alone. The last row is empty because alternating letters make mountains and valleys equal, and Maekawa asks for a difference of two.which sectors are strictly smallestletterings admittedno sector strictly smallest8one, at position 04two, opposite0two, adjacent2every sector0

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 order decides the count

Ask how many mountain-and-valley letterings a vertex admits and the answer looks as though it should depend on the angles. It does not. Three of the four conditions never see an angle at all, and the fourth asks only which sector is smallest — so the count is a function of a combinatorial arrangement, and a walk round the cycle that never looks at a vertex reproduces it exactly.

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