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Journal for Geometry and Graphics 21 (2017), No. 2, 169--178
Copyright Heldermann Verlag 2017



Enumeration of Flat-Foldable Crease Patterns in the Square/Diagonal Grid and Their Folded Shapes

Yoshihisa Matsukawa
Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennohdai, Tsukuba Ibaraki 305-8573, Japan
matsukawa@npal.cs.tsukuba.ac.jp

Yohei Yamamoto
Giken Ltd., 3948-1 Nunoshida, Kohchi 781-5195, Japan
yohey.yamamort@gmail.com

Jun Mitani
Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennohdai, Tsukuba, Ibaraki 305-8573, Japan
mitani@npal.cs.tsukuba.ac.jp



The crease patterns of several basic origami shapes fall within the 45-degree grid system, i.e., the square/diagonal grid of a 4 times 4 size. This grid system is easily creased on a square, and its flexibility allows to make a variety of shapes by folding along the edges. However, until now, the number of shapes made from a grid has not been known. We enumerated all possible formal crease patterns that are locally flat-foldable, along with their folded shapes, and found that we could enumerate 259,650,300 and 13,452 respectively. Further, we verified that all the shapes can be folded flat without any self-intersections. Some formal crease patterns that are not flat-foldable due to self-intersections were also found.

Keywords: Origami, crease pattern, flat-foldable, square/diagonal grid pattern.

MSC: 52B70; 68W05

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