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

TL;DR

This paper introduces an efficient method for approximately enumerating flat-foldings in single-vertex origami using a physical model and statistical mechanics, providing asymptotic predictions and insights into local stacking orders.

Contribution

It proposes a novel approach combining physical models and numerical methods for approximate enumeration of flat-foldings, with applications to random origami diagrams and size-dependent analysis.

Findings

Approximate enumeration results for flat-foldings of single-vertex origami.

Asymptotic size dependence and predictions for infinite size.

Insights into local stacking orders and forcing sets in origami folding.

Abstract

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.