Folding a paper strip made up of n congruent squares into the form of a single square is called strip folding. Strip folding has been studied from a computational and an applicational perspective as a subject in flat-folding origami. In this manuscript, we discern the algebraic structure beneath the folding behavior by defining two sorts of operators. We demonstrate how each operator cooperates with the other in a loosely defined “linear” manner. With all of the flatly folded states of the strips serving as its objects and composited operations serving as its morphisms, this algebraic structure also serves as a possible construction of a monoidal category. Our finding suggests  the possibility of employing some algebraic methodologies to strip folding. Conversely, strip folding can be used to visualize some abstract categorical concepts.

flat-folding origami, monoidal category.

Received: February 7, 2023; Accepted: March 17, 2023; Published: March 27, 2023

How to cite this article: Yiyang Jia and Jun Mitani, Making strip folding a monoidal category, JP Journal of Algebra, Number Theory and Applications 61(1) (2023), 1-18. http://dx.doi.org/10.17654/0972555523008

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

References:

[1] M. Bern and B. Hayes, The complexity of flat origami, Ann. ACM-SIAM Symposium on Discrete Algorithms, ACM, 1996, pp. 175-183.
[2] J. Justin, Mathematics of Origami, Part 9, British Origami, 1986, pp. 28-30.
[3] K. Kasahara and T. Takahama, Origami for the Connoisseur, New York, Japan Publications, 1987.
[4] 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(1) (2004), 23-46.
[5] H. A. Akitaya, K. C. Cheung, E. D. Demaine, T. Horiyama, T. C. Hull, J. S. Ku, T. Tachi and R. Uehara, Box Pleating is Hard, JCDCGG, Japan, 2016, pp. 167 179.
[6] R. Hartshorne, Algebraic Geometry, Berlin, New York, Springer-Verlag, 1977.
[7] E. D. Demaine, M. L. Demaine, J. S. B. Mitchell and T. Tachi, Continuous foldability of polygonal paper, Discrete Comput. Geom. 38(3) (2007), 453-463.
[8] Y. Jia, J. Mitani and R. Uehara, Logical matrix representations in map folding, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences 105(10) (2022), 1401-1412.
[9] Y. Jia and J. Mitani, Category of strip folding in terms of a Boolean matrix representation, JP Journal of Algebra, Number Theory and Applications 58 (2022), 19-36.
[10] C. Heunen and V. Jamie, Categories for Quantum Theory: An Introduction, Oxford University Press, 2019.
[11] R. I. Nishat, Map folding (Doctoral dissertation), 2013.
[12] A. Hora and N. Obata, Quantum Probability and Spectral Analysis of Graphs, Springer Science and Business Media, 2007.