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Maekawa's theorem is a theorem in the mathematics of paper folding named after Jun Maekawa. It relates to flat-foldable origami crease patterns and states that at every vertex, the numbers of valley and mountain folds always differ by two in either direction.[1] The same result was also discovered by Jacques Justin[2] and, even earlier, by S. Murata.[3]

Parity and coloring

One consequence of Maekawa's theorem is that the total number of folds at each vertex must be an even number. This implies (via a form of planar graph duality between Eulerian graphs and bipartite graphs) that, for any flat-foldable crease pattern, it is always possible to color the regions between the creases with two colors, such that each crease separates regions of differing colors.[4] The same result can also be seen by considering which side of the sheet of paper is uppermost in each region of the folded shape.
Related results

Maekawa's theorem does not completely characterize the flat-foldable vertices, because it only takes into account the numbers of folds of each type, and not their angles. Kawasaki's theorem gives a complementary condition on the angles between the folds at a vertex (regardless of which folds are mountain folds and which are valley folds) that is also necessary for a vertex to be flat-foldable.
References

Kasahara, K.; Takahama, T. (1987), Origami for the Connoisseur, New York: Japan Publications.
Justin, J. (June 1986), "Mathematics of origami, part 9", British Origami: 28–30.
Murata, S. (1966), "The theory of paper sculpture, II", Bulletin of Junior College of Art (in Japanese), 5: 29–37.

Hull, Thomas (1994), "On the mathematics of flat origamis" (PDF), Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994), Congressus Numerantium, 100, pp. 215–224, MR 1382321. See in particular Theorem 3.1 and Corollary 3.2.

Weisstein, Eric W. "Maekawa's Theorem". MathWorld.

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Mathematics of paper folding
Flat folding

Big-little-big lemma Crease pattern Huzita–Hatori axioms Kawasaki's theorem Maekawa's theorem Map folding Napkin folding problem Pureland origami Yoshizawa–Randlett system

Strip folding

Dragon curve Flexagon Möbius strip Regular paperfolding sequence

3d structures

Miura fold Modular origami Paper bag problem Rigid origami Sonobe

Polyhedra

Alexandrov's uniqueness theorem Flexible polyhedron (Bricard octahedron, Steffen's polyhedron) Net Star unfolding

Miscellaneous

Fold-and-cut theorem Lill's method

Publications

Geometric Exercises in Paper Folding Geometric Folding Algorithms A History of Folding in Mathematics Origami Polyhedra Design

People

Margherita Piazzola Beloch Erik Demaine Martin Demaine Britney Gallivan Rona Gurkewitz David A. Huffman Tom Hull Kôdi Husimi Humiaki Huzita Toshikazu Kawasaki Robert J. Lang Anna Lubiw Jun Maekawa Joseph O'Rourke