Folding Polyominoes into (Poly)Cubes

Abstract

We study the problem of folding a polyomino [Formula: see text] into a polycube [Formula: see text], allowing faces of [Formula: see text] to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of [Formula: see text] or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of [Formula: see text]), and (d) must lie on just the polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of [Formula: see text]. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.