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.

Highlights

  • The study of unfolding polyhedra was popularized by Albrecht Durer in the early 16th century in his influential book The Painter’s Manual

  • We explore ridge unfoldings of a convex polytope P by focusing on the combinatorics of the arrangement of its facets in the unfolding

  • The Roberts graph has been helpful in understanding the geometry of unfoldings through rolls, the language of chord diagrams, from the theory of Vassiliev knot and link invariants [2], is better suited to frame the study of spanning cycles

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Summary

Introduction

The study of unfolding polyhedra was popularized by Albrecht Durer in the early 16th century in his influential book The Painter’s Manual. Shephard [17] conjectures that every convex polyhedron can be cut along certain edges and admits a net This claim remains tantalizingly open [9], resulting in numerous areas of exploration [6, 8, 11, 13] and influencing book manuscripts [7, 15]. In 1984, Turney [19] enumerates the 261 ridge unfoldings of the 4-cube, and in 1998, Buekenhout and Parker [3] extend this to the other five regular convex 4-polytopes Both of these works focus on combinatorial rather than geometric unfolding results. A decade later, Miller and Pak [14] construct an algorithm which provides an unfolding of polytopes without overlap Their method allows cuts interior to facets, not just along ridges.

Duality
Unfolding Algorithm
Net Guarantee
Bounding Boxes
Token Sliding Game
Path Nets
Chord Diagrams
Enumerations
Conclusion

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