The count halves the moment the tie is broken
sector-ties 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
view: "sweep", from: 30, to: 89.9
view: "sweep", from: 45, to: 89.9, samples: 60
view: "vertex", a: 45
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.
- along the whole family whose two smallest sectors are equal the count of foldable assignments is 8, at every angle from 30° to 89.9° ×3
- a degree-four vertex whose two smallest sectors are equal admits 8 flat-foldable assignments and one whose smallest is strictly smallest admits 4 ×1
- a vertex a tenth of a degree away from a tie admits 4 — the count halves the moment the tie is broken, however small the break ×1
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 tie is not a decision
The crimp reduction decides a vertex by folding its smallest sector away, and where two sectors tie for smallest it has no forced move and must try each of them. That search is not rare — on the vertex at the centre of the first base anybody folds it happens for fourteen of the sixteen letterings — and it has never once changed the answer.
The dial that decides nothing
Turn a twist tessellation's angle from one fence to the other and every measurable thing about it changes: the smallest sector goes from 88 degrees to under one, the pleats swallow a quarter of the sheet and then almost none of it, the folded footprint changes by a third. The number of ways it can be creased does not change at all — sixteen, at every angle tested — because the lemma reads which sector is smallest and never how small.
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.
Walking between two foldings
The letterings a vertex folds in are always counted and never navigated. Counting says a generic degree-six vertex has eight of them; navigating says that changing any two creases turns any one into any other, and that changing two neighbouring creases does not — and that the vertices which come apart are the ones with no coincidences in them, which is the opposite of what every other measurement here would suggest.
Where the lemma says nothing
The big-little-big lemma asks for a sector strictly smaller than both its neighbours, and the word doing the work is strictly. At a vertex whose two smallest sectors are equal the lemma has no opinion at all — and those are the vertices origami actually uses. The count of markings the conditions admit doubles, discontinuously, at exactly the angles everybody folds.
All the figures · The flat-folding field · The patterns a reader can fold