Efficient Enumeration of Flat-Foldable Single Vertex Crease Patterns

Abstract

We investigate enumeration of distinct flat-foldable crease patterns with natural assumptions. Precisely, for a given positive integer n, potential set of n crease lines are incident to the center of a sheet of disk paper at regular angles. That is, every angle between adjacent lines is equal to $2\pi /n$. Then each line is assigned one of “mountain,” “valley,” and “flat (or consequently unfolded).” That is, we enumerate all flat-foldable crease patterns with up to n crease lines of unit angle $2\pi /n$. We note that two crease patterns are equivalent if they are equal up to rotation and reflection. In computational origami, there are two well-known theorems for flat-foldability: the Kawasaki Theorem and the Maekawa Theorem. The first one is a necessary and sufficient condition of crease layout, however, it does not give us valid mountain/valley assignments. The second one is a necessary condition between the number of “mountain” and that of “valley.” However, sufficient condition(s) is(are) not known. Therefore, we have to enumerate and check flat-foldability one by one using other algorithm. In this research, we develop the first algorithm for the above stated problem by combining these results in a nontrivial way, and show its analysis of efficiency. We also give experimental results, which give us a new series of integer sequence.