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We enumerate all convex rectangles that each of them consists of two unit angle triangles connected on edges. Such convex rectangles often appear in origami designs. We also generalize our enumeration problem with input a pair of n for \(\pi /n\) unit angle and m for m-gons to be enumerated: We use \(m - 2\) triangles to build an m-gon such that the triangles can be a result of triangulation of the m-gon. In this paper, we observe the similarity between the problem and the silhouette puzzles, and construct an enumeration algorithm by modifying the algorithm for Nana-kin-san puzzle. The numbers of enumerated rectangles are 78 and 497 for \(n=8\) and \(n=12\) , respectively. We show the rectangles for \(n=8\) in the appendix with the hope that it will be used as a catalog of shapes for origami designs.

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Efficient Enumeration of Rectangles in Origami Design

  • 会议论文

  • 发表日期：

    2026年

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  • K. Ouchi,
  • H. Komatsu,
  • R. Uehara

摘要

We enumerate all convex rectangles that each of them consists of two unit angle triangles connected on edges. Such convex rectangles often appear in origami designs. We also generalize our enumeration problem with input a pair of n for \(\pi /n\) unit angle and m for m-gons to be enumerated: We use \(m - 2\) triangles to build an m-gon such that the triangles can be a result of triangulation of the m-gon. In this paper, we observe the similarity between the problem and the silhouette puzzles, and construct an enumeration algorithm by modifying the algorithm for Nana-kin-san puzzle. The numbers of enumerated rectangles are 78 and 497 for \(n=8\) and \(n=12\) , respectively. We show the rectangles for \(n=8\) in the appendix with the hope that it will be used as a catalog of shapes for origami designs.