Folding polyiamonds into octahedra

We study polyiamonds (polygons arising from the triangular grid) that fold into smallest yet unstudied platonic solid – octahedron. show a number of results. Firstly, we characterize foldable containing hole positive area, namely each but one polyiamond is foldable. Secondly, convex folds octahedron if and only it contains five polyiamonds. thirdly present sharp size bound: While there exist unfoldable 14, every at least 15 This clearly implies can test in polynomial time whether given Lastly, for any assignment integers to faces, exists such triangles covering face equal assigned number.