Summary: A parallelohedron is a convex polyhedron which tiles 3-dimensional space by translations only. A polyhedron \(\sigma\) is said to be an atom for the set \(\Pi\) of parallelohedra if for each parallelohedron \(P\) in \(\Pi\), there exists an affine-stretching transformation \(A:\mathbb{R}^{3} \to \mathbb{R}^{3}\) such that \(A(P)\) is the union of a finite number of copies of \(\sigma\). In this paper, we present two different atoms for the parallelohedra, and determine the number of these atoms used to make up each parallelohedron. We also show an arrangement of the parallelohedra in lattice-like order and introduce the notion of indecomposability.
For the entire collection see [Zbl 1282.05006].