Parallelohedra, old and new

Abstract

It is shown that from a skewed, skeletal (edges and vertices), truncated octahedron, skewed skeletons can be derived of the other four convex parallelohedra found by Fedorov in 1885. In addition, three new nonconvex parallelohedra are produced, a counterexample to a statement by Grünbaum. This opens several new ways to view atomic positions in crystals, and new avenues in geometry.