A theorem giving a criterion for an origami construction to be flat. Kawasaki's theorem states that a given
crease pattern can be folded to a flat origamiiff all the sequences of angles , ..., surrounding each (interior) vertex fulfil the following
condition
Note that the number of angles is always even; each of them corresponds to a layer of the folded sheet.
The rule evidently applies to the case of a rectangular sheet of paper folded twice, where the crease pattern is formed by the bisectors. But there are many more interesting examples where the above property can be checked (see, for example, the crane origami in the above figure).
Andersen, E. M. "Origami and Math." http://www.paperfolding.com/math/.Bern, M. and Hayes, B. "The Complexity of Flat Origami." In Proceedings of
the 7th Annual ACM-SIAM Symposium on Discrete Algorithms. Atlanta, GA, pp. 175-183,
1996.Demaine, E. D. Folding and Unfolding. Doctoral Thesis,
University of Waterloo, Canada, p. 26, 2001. http://etd.uwaterloo.ca/etd/eddemaine2001.pdf.Hull,
T. "On the Mathematics of Flat Origamis." Congr. Numer.100,
215-224, 1994.Hull, T. "MA 323A Combinatorial Geometry!: Notes
on Flat Folding." http://web.merrimack.edu/hullt/combgeom/flatfold/flat.html.Justin,
J. "Towards a Mathematical Theory of Origami." In Proceedings of the
2nd International Meeting of Origami Science and Scientific Origami (Ed. K. Miura).
Otsu, Japan, pp. 15-29, 1994.Kawasaki, T. "On the Relation
Between Mountain-Creases and Valley-Creases of a Flat Origami." In Proceedings
of the 1st International Meeting on Origami Science and Technology (Ed. H. Huzita).
Ferrara, Italy, pp. 229-237, 1989.
, ..., 
surrounding each (interior) vertex fulfil the following
condition
