Edge-Unfolding Polycubes with Orthogonally Convex Layers

Abstract

A polycube is an orthogonal polyhedron composed of unit cubes glued together along entire faces, homeomorphic to a sphere. A polycube layer is the section of the polycube that lies between two horizontal cross-sections of the polycube at unit distance from each other. An edge unfolding of a polycube involves cutting its surface along any of the constituent cube edges and flattening it into a single, non-overlapping planar piece. We show that any polycube with orthogonally convex layers can be edge unfolded.