Kawasaki's theorem

Kawasaki's theorem states that a single-vertex crease pattern is locally flat-foldable only if its alternating sector angles sum to 180 degrees; for nondegenerate patterns, this condition is also sufficient.

Sector angle: An angle between two consecutive crease rays meeting at a vertex. The theorem adds alternate sector angles around one vertex.
Origami: The art and mathematics of folding sheets, traditionally paper, into forms. Kawasaki's theorem is a local geometric result about paper folding.
Toshikazu Kawasaki: A Japanese mathematician known for work on origami geometry and the theorem bearing his name. His formulation established the alternating-angle criterion for flat-foldable vertices.
Miura-ori: A repeated parallelogram crease pattern that folds into a compact, deployable sheet. Its repeated vertices illustrate how local angle constraints support a foldable pattern.
Global flat-foldability: The ability of an entire crease pattern to fold flat while satisfying all geometric and physical constraints. Local satisfaction of Kawasaki's condition does not guarantee a globally flat-foldable sheet.
Crease pattern: A diagram of lines along which a sheet is intended to be folded. Kawasaki's condition tests local flat-foldability of a crease pattern.
Vertex: A point where two or more lines, edges, or surfaces meet. The theorem concerns crease rays meeting at a single vertex.
Jun Maekawa: A Japanese origami theorist associated with a rule for mountain and valley crease counts. His local crease-count rule developed alongside Kawasaki's angle condition.
Origami engineering: The use of folding principles to design deployable structures and functional devices. Engineers use local foldability tests when designing crease-based structures.
Self-intersection: The crossing or overlap of parts of a geometric object with itself. A locally valid fold may still cause distant parts of the sheet to collide.

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