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.

Voir la notice de l'article provenant de la source Heldermann Verlag

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},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2017},
     url = {http://geodesic.mathdoc.fr/item/JGG_2017_21_2_JGG_2017_21_2_a2/}
}
TY  - JOUR
AU  - Y. Matsukawa
AU  - Y. Yamamoto
AU  - J. Mitani 
TI  - Enumeration of Flat-Foldable Crease Patterns in the Square/Diagonal Grid and Their Folded Shapes
JO  - Journal for geometry and graphics
PY  - 2017
SP  - 169
EP  - 178
VL  - 21
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JGG_2017_21_2_JGG_2017_21_2_a2/
ID  - JGG_2017_21_2_JGG_2017_21_2_a2
ER  - 
%0 Journal Article
%A Y. Matsukawa
%A Y. Yamamoto
%A J. Mitani 
%T Enumeration of Flat-Foldable Crease Patterns in the Square/Diagonal Grid and Their Folded Shapes
%J Journal for geometry and graphics
%D 2017
%P 169-178
%V 21
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JGG_2017_21_2_JGG_2017_21_2_a2/
%F JGG_2017_21_2_JGG_2017_21_2_a2
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/