Morphing polyhedra with parallel faces: Counterexamples

Two simple polyhedra P and Q (not necessarily convex) are parallel if they share the same edge graph G and each face of P has the same outward-facing unit normal as the corresponding face in Q. Parallel polyhedra P and Q admit a parallel morph if the vertices can be moved in a continuous manner taking us from P to Q such that at all times the intermediate polyhedron determined by the vertex configuration and graph G is both simple and parallel with P (and Q). In this note, we show that even for very restrictive classes of orthogonal polyhedra, there exist parallel polyhedra that do not admit a parallel morph.