An Efficient Enumeration of Flat-Foldings : Study on Random Single Vertex Origami

This paper deals with themes such as approximate counting/evaluation of the total number of flat-foldings for random origami diagrams, evaluation of the values averaged over various instances, obtaining forcing sets for general origami diagrams, and evaluation of average computational complexity. An approach to the above problems using a physical model and an efficient size reduction method for them is proposed. Using a statistical mechanics model and a numerical method of approximate enumeration based on it, we give the result of approximate enumeration of the total number of flat-foldings of single-vertex origami diagram with random width of angles gathering around the central vertex, and obtain its size dependence for an asymptotic prediction towards the limit of infinite size. In addition, an outlook with respect to the chained determination of local stacking orders of facets caused by the constraint that prohibits the penetration of them is also provided from the viewpoint of organizing the terms included in the physical model. A method to efficiently solve the problem of the determination or enumeration of flat-foldings is discussed based on the above perspectives. This is thought to be closely related to forcing sets.
Comment: 16 pages, 11 figures. Submitted to Origami8: Proceedings of the 8th International Meeting on Origami in Science, Mathematics and Education (8OSME), Melbourne, August 16-18, 2024