WALCOM 2019: WALCOM: Algorithms and Computation pp 135-147

# Flat-Foldability for 1 × n Maps with Square/Diagonal Grid Patterns

• Yiyang Jia
• Yoshihiro Kanamori
• Jun Mitani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)

## Abstract

In this paper, we propose three conclusions for 1 × n maps with square/diagonal grid patterns. First, for a 1 × n map consisting of all the vertical creases and all the diagonal creases as well as a mountain-valley assignment, if it obeys the local flat-foldability, then it can always be globally flat-folded and one of its flat-folded state can be reached in O(n) time. Second, for a 1 × n map consisting of only square/diagonal grid pattern, it also can always be globally flat-folded and one of its flat-foldable state can be reached in O(n) time. We give theoretical proofs for both of them and propose corresponding algorithms. Then, we prove the NP-hardness of the problem of determining the global flat-foldability for a 1 × n map consisting of a square/diagonal grid pattern and a specific mountain-valley assignment. Also, we show that given an order of the faces for an m × n map with all the vertical creases and all the diagonal creases assigned to be mountains or valleys, we can determine its validity in O(mn) time.

## Keywords

Square/diagonal grid patterns Flat-foldability NP-hardness

## References

1. 1.
Bern, M., Hayes, B.: The complexity of flat origami. In: Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms SODA 1996, pp. 175–183. Society for Industrial and Applied Mathematics, Philadelphia (1996)Google Scholar
2. 2.
Akitaya, H.A., et al.: Box pleating is hard. In: Akiyama, J., Ito, H., Sakai, T. (eds.) JCDCGG 2015. LNCS, vol. 9943, pp. 167–179. Springer, Cham (2016).
3. 3.
Demaine, E., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, and Polyhedra. Cambridge University Press, Cambridge (2007)
4. 4.
Arkin, E.M., et al.: When can you fold a map? Comput. Geom. Theor. Appl. 29(1), 23–46 (2004)
5. 5.
Morgan, T.: Map folding. Master’s thesis, Massachusetts Institute of Technology (2012)Google Scholar
6. 6.
Nishat, R.I., Whitesides, S.: Canadian Conference on Computational Geometry, pp. 49–54 (2013)Google Scholar
7. 7.
Justin, J.: Aspects mathematiques du pliage de papier (mathematical aspects of paper fold). In: Huzita, H. (ed.) 1st International Meeting of Origami Science and Scientific Origami, pp. 263–277 (1989)Google Scholar
8. 8.
Hull, T.C.: Counting mountain-valley assignments for flat folds. arXiv preprint arXiv:1410.5022 (2014)
9. 9.
Matsukawa, Y., Yamamoto, Y., Mitani, J.: Enumeration of flat-foldable crease patterns in the square/diagonal grid and their folded shapes. J. Geom. Graph. 21(2), 169–178 (2017)
10. 10.
Kasahara, K., Takahama, T.: Origami for the Connoisseur. Japan Publications Inc. (1978)Google Scholar
11. 11.
Koehler, J.E.: Folding a strip of stamps. J. Combinatorial Theor. 5(2), 135–152 (1968)