Enumeration of crease patterns that satisfy locally flat-foldable conditions and solve the hole problem

Abstract

The hole problem in origami is, given a crease pattern with an area with the creases missing in it, a problem to consider how to fill in the area with suitable creases. An algorithm for solving the hole problem is known. However, crease patterns output by the algorithm have not been enumerated. In addition, when flat-foldable origami is targeted, it is not guaranteed that the crease patterns output by the algorithm are capable mountain-valley assignments that satisfy conditions of local flat-foldability (valid MV assignments). In our study, we have enumerated crease patterns output by the algorithm, and have implemented a function to determine whether they have valid MV assignments. As a result, we have enumerated crease patterns output by inputting polygons with four and six vertices and have made it possible to determine whether those crease patterns have valid MV assignments.