Origami & Folding
what one sheet can do
Concept

Sector — where it appears

The wedge of paper between two consecutive creases at a vertex. The sectors are what the angle conditions are about, and their sizes are the only geometry a local test of flat-foldability ever sees.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

MVMM40°foldsopposite across the small sectorMMVM40°does not foldthe same on both sidesboth satisfy Kawasaki and Maekawa — the angles and the counts are identical

The smallest sector decides

Two assignments can satisfy both flat-folding theorems and only one of them folds. What separates them is a condition about the smallest angle, and it is the first rule in the subject that is not about counting.

flat-folding · Flat-foldability 102°60°78°120°one smallest sectorthe lemma constrains one pair4 foldable assignmentsof the 16 markings120°60°60°120°two smallest sectors equalthe lemma constrains nothing8 foldable assignmentsof the 16 markings

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.

flat-folding · Sector angles

Named alongside it

The objects these essays reach for when they reach for this one.

AssignmentThe big-little-big lemmaNecessary conditionCollisionCountingDegeneracyGenericitySector angles

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