Unfolding cubes: nets, packings, partitions, chords

TL;DR

This paper proves that all ridge unfoldings of an n-cube produce non-overlapping nets and introduces combinatorial methods to classify their geometries and unfoldings.

Contribution

It establishes that every ridge unfolding of an n-cube is overlap-free and develops combinatorial frameworks for analyzing cube nets and their geometries.

Findings

All ridge unfoldings are overlap-free

Classification of cube nets via integer partitions

Analysis of path unfoldings using chord diagrams

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.