Efficient Enumeration of Flat-Foldable Single Vertex Crease Patterns

• Published in: IEICE Transactions on Information and Systems Vol. E102.D; no. 3; pp. 416 - 422
• Main Authors: OUCHI, Koji, UEHARA, Ryuhei
• Format: Journal Article
• Language: English
• Published: Tokyo The Institute of Electronics, Information and Communication Engineers 01.03.2019; Japan Science and Technology Agency
• Subjects: Algorithms; computational origami; Enumeration; enumeration algorithm; flat foldability; Kawasaki theorem; Maekawa theorem; Mountains; Theorems
• ISSN: 0916-8532; 1745-1361
• DOI: 10.1587/transinf.2018FCP0004

We investigate enumeration of distinct flat-foldable crease patterns under the following assumptions: positive integer n is given; every pattern is composed of n lines incident to the center of a sheet of paper; every angle between adjacent lines is equal to 2π/n; every line is assigned one of “mountain,” “valley,” and “flat (or consequently unfolded)”; crease patterns are considered to be equivalent if they are equal up to rotation and reflection. In this natural problem, we can use two well-known theorems for flat-foldability: the Kawasaki Theorem and the Maekawa Theorem in computational origami. Unfortunately, however, they are not enough to characterize all flat-foldable crease patterns. Therefore, so far, we have to enumerate and check flat-foldability one by one using computer. In this study, we develop the first algorithm for the above stated problem by combining these results in a nontrivial way and show its analysis of efficiency.