Unfolding Cubes: Nets, Packings, Partitions, Chords

Abstract

We show that every ridge unfolding of an $n$-cube is without self-overlap, yielding a valid net.  The results are obtained by developing machinery that translates cube unfolding into combinatorial frameworks.  Moreover, the geometry of the bounding boxes of these cube nets are classified using integer partitions, as well as the combinatorics of  path unfoldings seen through the lens of chord diagrams.