Geodesic
Enumeration of Flat-Foldable Crease Patterns in the Square/Diagonal Grid and Their Folded Shapes
Journal for geometry and graphics, Tome 21 (2017) no. 2, pp. 169-178 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 52B70, 68W05
Mots-clés : Origami, crease pattern, flat-foldable, square/diagonal grid pattern 
@article{JGG_2017_21_2_JGG_2017_21_2_a2,
     author = {Y. Matsukawa and Y. Yamamoto and J. Mitani },
     title = {Enumeration of {Flat-Foldable} {Crease} {Patterns} in the {Square/Diagonal} {Grid} and {Their} {Folded} {Shapes}},
     journal = {Journal for geometry and graphics},
     pages = {169--178},
     year = {2017},
     volume = {21},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JGG_2017_21_2_JGG_2017_21_2_a2/}
}
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Y. Matsukawa; Y. Yamamoto; J. Mitani . Enumeration of Flat-Foldable Crease Patterns in the Square/Diagonal Grid and Their Folded Shapes. Journal for geometry and graphics, Tome 21 (2017) no. 2, pp. 169-178. http://geodesic.mathdoc.fr/item/JGG_2017_21_2_JGG_2017_21_2_a2/