  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.
