Smallest polyhedral tilings of 3-tori by parallelohedra (Q2022357)

A parallelohedron is a convex polytope that admits a face-to-face tiling of the Euclidean space by translations. There are two parallelohedra in dimension two and five parallelohedra in dimension three. An \(n\)-torus is the quotient of the \(n\)-dimensional Euclidean space by a discrete translation group isomorphic with \(\mathbb Z^n\). Smallest polyhedral tilings of the \(n\)-torus are such tilings that identifications on the boundary of tiles do not occur and any two tiles have a nonempty intersection. For 2-tori the three minimal models are well known. In the paper the five 3-dimensional parallelohedra are investigated. Each of them admits a smallest polyhedral tiling of 3-torus, moreover for three parallelohedra this is unique up to natural equivalence. Altogether 14 inequivalent minimal models are obtained.