Unfoldings and nets of regular polytopes

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 their higher-dimensional analogs, the regular polytopes. Three classes of regular polytopes exist for all dimensions (n-simplex, n-cube, n-orthoplex) and three additional regular polytopes appear only in four-dimensions (24-cell, 120-cell, 600-cell). It was recently proven that all unfoldings of the n-cube yield nets. We extend this to the n-simplex and the 4-orthoplex using the geometry of simplicial chains. Finally, we demonstrate failure of this property for any orthoplex of higher dimension, as well as for the 600-cell, providing counterexamples. We conjecture failure for the two remaining open cases, the 24-cell and the 120-cell.