Unfoldings and Nets of Regular Polytopes

TL;DR

This paper investigates whether unfoldings of regular polytopes in arbitrary dimensions produce non-overlapping nets, confirming this for some polytopes like the n-cube and n-simplex, but revealing failures for higher-dimensional orthoplexes.

Contribution

It extends the understanding of unfoldings of regular polytopes to higher dimensions, identifying which polytopes always produce nets and where overlaps occur.

Findings

All unfoldings of the n-cube yield nets.

Unfoldings of the n-simplex also yield nets.

Higher-dimensional orthoplexes can produce overlaps.

Abstract

Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for regular polytopes in arbitrary dimensions, notably the simplex, cube, and orthoplex. It was recently proven that all unfoldings of the $n$-cube yield nets. We show this is also true for the $n$-simplex and the $4$-orthoplex but demonstrate its surprising failure for any orthoplex of higher dimension.