It is unknown whether every polycube (polyhedron constructed by gluing cubes face-to-face) has an edge unfolding, that is, cuts along edges of the cubes that unfolds the polycube to a single nonoverlapping polygon in the plane. Here we construct polycubes that have no edge zipper unfolding where the cut edges are further restricted to form a path.
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Demaine, Erik D.; Demaine, Martin L.; Eppstein, David; and O'Rourke, Joseph, "Some Polycubes Have No Edge Zipper Unfolding" (2020). Computer Science: Faculty Publications, Smith College, Northampton, MA.