Kawasaki's theorem

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 2"n" in a single vertex fold and let α₁, α₂, ⋯, α_2"n" be the consecutive angles between the creases. Then then v is a flat vertex fold if and only if

: $alpha\; \_1\; -\; alpha\; \_2\; +\; alpha\; \_3\; -\; alpha\; \_4\; +\; cdots\; -\; alpha\; \_\{2n\}\; =\; 0.,$

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.

References

Hull, T. "MA 323A Combinatorial Geometry!: Notes on Flat Folding." [http://www.merrimack.edu/~thull/combgeom/flatfold/flat.html] .

Good Origami and Math Websites

www.paperfolding.com/math