Efficient Enumeration of Flat-Foldable Single Vertex Crease Patterns
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.
KeywordsBinary String Single Vertex Linear Time Algorithm Enumeration Algorithm Efficient Enumeration
We would like to thank Yota Otachi for his fruitful discussions and comments. This work is partially supported by MEXT/JSPS Kakenhi Grant Number 26330009 and 24106004.
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