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- W4288804594 abstract "We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid -- the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of positive area, namely each but one polyiamond is foldable. Secondly, we show that a convex polyiamond folds into the octahedron if and only if it contains one of five polyiamonds. We thirdly present a sharp size bound: While there exist unfoldable polyiamonds of size 14, every polyiamond of size at least 15 folds into the octahedron. This clearly implies that one can test in polynomial time whether a given polyiamond folds into the octahedron. Lastly, we show that for any assignment of positive integers to the faces, there exist a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number." @default.
- W4288804594 created "2022-07-30" @default.
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- W4288804594 date "2022-07-28" @default.
- W4288804594 modified "2023-09-28" @default.
- W4288804594 title "Folding Polyiamonds into Octahedra" @default.
- W4288804594 doi "https://doi.org/10.48550/arxiv.2207.13989" @default.
- W4288804594 hasPublicationYear "2022" @default.
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