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Kawasaki's theorem is a theorem in the mathematics of paper folding that gives a criterion for whether a given crease pattern is locally flat-foldable.

Statement of the theorem

Let v be a vertex of degree 2n in a single vertex fold and let α₁, α₂, ⋯, α_2n be the consecutive angles between the creases. Then v is a flat vertex fold if and only if

That is basically saying, if you take the angle measurement of every other angle around a point and add them up, the sum will be 180 degrees.

References

• Hull, T. "MA 323A Combinatorial Geometry!: Notes on Flat Folding." [1]