Overlapping of Lattice Unfolding for Cuboids

Abstract

A polygon obtained by cutting the surface of a polyhedron is called an unfolding. An unfolding obtained by cutting along only edges is called an edge unfolding. An unfolding may have overlapping, which are self-intersections on its boundary. It is a well-known open question in computational origami whether or not every convex polyhedron has a non-overlapping edge unfolding. On the other hand, Sharir and Schorr showed that any convex polyhedron could unfold without overlapping when allowed to cut its faces. Therefore, there is a gap between edge unfoldings and general unfoldings. Bridging this gap is necessary as a foothold on this open question of edge unfolding. Instead of cutting faces arbitrarily, there are studies considering whether specific cutting lines on the faces can result in unfoldings without overlaps. Lattice unfoldings of a cuboid made by unit cubes are one such example. A lattice unfolding of a cuboid is a polygon obtained by cutting the faces along the edges of unit squares. An unfolding may have overlapping, even in the case of small cuboids. In particular, Uno showed that a 1 × 1 × 3-cuboid has an overlapping lattice unfolding, while Mitani and Uehara showed the same for three faces of a 1 × 2 × 3-cuboid. In contrast, it is known that some cuboids have no overlapping lattice unfolding. Hearn showed it for a 1 × 1 × 2-cuboid, and Sugihara showed the same for a 2 × 2 × 2-cuboid. In this study, we completely clarify the existence of overlapping lattice unfoldings, which also contains the case where the sizes are non-integers.