Unfoldings of doubly covered polyhedra and applications to space-fillers

We study unfoldings (developments) of doubly covered polyhedra, which are space-fillers in the case of cuboids and some others. All five types of parallelohedra are examples of unfoldings of doubly covered cuboids (Proposition 1). We give geometric properties of convex unfoldings of doubly covered cuboids and determine all convex unfoldings (Theorem 1). We prove that every unfolding of doubly covered cuboids has a space-filling (consisting of its congruent copies) generated by three specified translates and three specified rotations, and that all such space-fillers are derived from unfoldings of doubly covered cuboids (Theorem 2). Finally, we extend these results from cuboids to polyhedra which are fundamental regions of the Coxeter groups generated by reflections in the 3-space and which have no obtuse dihedral angles (Theorem 3).