Folding polyiamonds into octahedra

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 exists a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number.