Overlapping edge unfoldings for convex regular-faced polyhedra

Herein, we discuss the existence of overlapping edge unfoldings for convex regular-faced polyhedra. Horiyama and Shoji showed that there are no overlapping edge unfoldings for all platonic solids and five of the Archimedean solids. The remaining five Archimedean solids were found to have edge unfoldings that overlap. In this study, we propose a method called rotational unfolding to find an overlapping edge unfolding of a polyhedron. We show that all the edge unfoldings of an icosidodecahedron, a rhombitruncated cuboctahedron, an n-gonal Archimedean prism (3≤n≤23), an m-gonal Archimedean antiprism (3≤m≤11), and 48 types of Johnson solids do not overlap. Our algorithm finds overlapping edge unfoldings for the snub cube, and 44 types of Johnson solids. We present analytic proof that an overlapping edge unfolding exists in an n-gonal Archimedean prism (n≥24), and an m-gonal Archimedean antiprism (m≥12). Our results prove the existence of overlapping edge unfoldings for convex regular-faced polyhedra.