This study utilizes category theory to enhance the understanding of flat-fold origami, focusing on the structural characteristics and representation of self-intersections within categorical definitions. We introduce the category of flat states and its skeleton category  based on permutation equivalence. We explored the localized subcategories  and  besides establishing a Heyting algebra structure on  proved its lack of a locale structure, confirming that  functions as a -topos but does not qualify as a Grothendieck topos. These findings categorize the domain of flat-fold origami as a category of -topoi, offering significant theoretical insights for analyzing flat-fold origami.

Heyting algebra, origami, flat-fold.

Received: May 28, 2024; Accepted: July 3, 2024; Published: July 10, 2024

How to cite this article: Yiyang Jia and Jun Mitani, Heyting algebra in flat origami, JP Journal of Algebra, Number Theory and Applications 63(5) (2024), 383-396. https://doi.org/10.17654/0972555524023

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] Hugo A. Akitaya, Kenneth C. Cheung, Erik D. Demaine, Takashi Horiyama, Thomas C. Hull, Jason S. Ku, Tomohiro Tachi and Ryuhei Uehara, Box pleating is hard, Japanese Conference on Discrete and Computational Geometry and Graphs, Springer, 2015, pp. 167-179.
[2] E. M. Arkin, M. A. Bender, E. D. Demaine, M. L. Demaine, J. S. B. Mitchell, S. Sethia and S. S. Skiena, When can you fold a map? Comput. Geom. 29 (2002), 23-46.
[3] M. Bern and B. Hayes, The complexity of flat origami, Ann. ACM-SIAM Symposium on Discrete Algorithms, ACM, 1996, pp. 175-183.
[4] Thomas Hull, The combinatorics of flat folds: a survey, Origami3: Proceedings of the 3rd International Meeting of Origami Science, Math, and Education, 2002, pp. 29-38.
[5] Yiyang Jia and Jun Mitani, Order theory in strip folding, JP Journal of Algebra, Number Theory and Applications 62(1) (2023), 13-34.
[6] Yiyang Jia, Jun Mitani and Ryuhei Uehara, Efficient algorithm for map folding with a box-pleated crease pattern, Journal of Information Processing 28 (2020), 806-815.
[7] R. I. Nishat, Map folding, Master Thesis, University of Victoria, 2013.
[8] Ryuhei Uehara, Stamp foldings with a given mountain-valley assignment, Origami5: Proceedings of the 5th International Meeting of Origami Science, Mathematics and Education (AK Peters/CRC Press, 2011), 2011, pp. 585-597.