Kawasaki's Theorem

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 origami iff all the sequences of angles alpha_1, ..., alpha_(2n) surrounding each (interior) vertex fulfil the following condition

 alpha_1+alpha_3+...+alpha_(2n-1)=alpha_2+alpha_4+...+alpha_(2n)=180 degrees.

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).

See also

Flat Origami, Maekawa's Theorem, Origami

This entry contributed by Margherita Barile

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Andersen, E. M. "Origami and Math.", 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., T. "On the Mathematics of Flat Origamis." Congr. Numer. 100, 215-224, 1994.Hull, T. "MA 323A Combinatorial Geometry!: Notes on Flat Folding.", 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.

Referenced on Wolfram|Alpha

Kawasaki's Theorem

Cite this as:

Barile, Margherita. "Kawasaki's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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