2022/07/28 by Eva Stehr, Linda Kleist, Stehr, Eva +1
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · #Advanced Materials and Mechanics #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Geometric and Algebraic Topology #Microtubule and mitosis dynamics
paper · pdf · doi:10.48550/arxiv.2207.13989
openalex publication_date 2022/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.