Maekawa's theorem

Maekawa's theorem states that at a flat-foldable origami vertex, the numbers of mountain and valley folds differ by exactly two.

Origami vertex: A point where two or more creases meet in an origami crease pattern. Maekawa's theorem applies locally at this meeting point.
Kawasaki's theorem: A theorem stating that a single-vertex crease pattern is flat-foldable only if alternating sector angles sum to 180 degrees. It supplies the complementary angle condition for flat-foldability at a vertex.
Four-crease vertex: A crease pattern with four creases meeting at a single point. A flat-foldable example must assign three creases one way and one the other.
Origami mathematics: The mathematical study of folding, crease patterns, and the geometry of paper structures. Maekawa's theorem is one local constraint within this broader subject.
Crease pattern: A diagram of the creases and boundaries formed by folding or unfolding a sheet. The theorem constrains mountain and valley assignments within this diagram.
Fold angle: The angle between the two faces adjacent to a crease after folding. Mountain and valley assignments determine the direction in which crease angles close.
Six-crease vertex: A crease pattern with six creases meeting at a single point. Its flat-foldable assignments must split into four mountain and two valley folds, or the reverse.
Flat-foldability problem: The problem of determining whether a crease pattern can be folded flat without collisions. Satisfying Maekawa's rule alone does not solve this broader feasibility problem.
Mountain fold: A crease that folds so the paper's surface forms a ridge facing outward. Mountain folds are one of the two counts whose difference the theorem fixes.
Rigid origami: The study of origami folds that move while each paper face remains rigid. Maekawa's count condition does not by itself guarantee a continuous rigid-folding motion.

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