On Reversibility among Parallelohedra

Abstract

Given two convex polyhedra α and β, we say that α and β are a reversible pair if α has a dissection into a finite number of pieces which can be rearranged to form β in such a way that no face of the dissection of α includes any part of an edge of α, no face of the dissection of β includes any part of an edge of β, the pieces are hinged on some of their edges so that the pieces of the dissection are connected as in a tree-structure, all of the exterior surface of α is in the interior of β, and all of the exterior surface of β comes from the interior of α. Let $\mathfrak{P}_{i}$ denote one of the five families of parallelohedra (see Section 2 for the corresponding definitions). In this paper, it is shown that given an arbitrary canonical parallelohedron P, there exists a canonical parallelohedron $Q \in \mathfrak{P}_{i}$ such that the pair P and Q is reversible for each $\mathfrak{P}_{i}$.