Some Polycubes Have No Edge Zipper Unfolding

TL;DR

This paper investigates the edge unfolding problem for polycubes and constructs examples that lack an edge zipper unfolding, where cuts form a path, highlighting limitations in unfolding methods.

Contribution

It demonstrates the existence of polycubes without edge zipper unfoldings, advancing understanding of polycube unfolding limitations.

Findings

Certain polycubes have no edge zipper unfolding.

Edge zipper unfoldings are not always possible for all polycubes.

The work provides explicit counterexamples.

Abstract

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.