Characterization of the three-dimensional multiple translative tiles
Let \(X\) be a discrete multiset in the three-dimensional Euclidean space and let \(k\) be a positive integer. \(T+X\) is called a \(k\)-fold translative tiling of \(\mathbb{E}^n\) and \(T\) is called a \(k\)-fold translative tile if every point of \(\mathbb{E}^n\) belongs to at least \(k\) translates of \(T\) in \(T+X\) and belongs to at most \(k\) translates of \(\mathrm{int}(T)\) in \(\mathrm{int}(T)+X\). It is known that any \(k\)-fold translative tile must be a convex polytope. \par The paper proves that in the three-dimensional Euclidean space, every \(k\)-fold translative tile for \(k=2\), 3 and 4 is a parallelohedron, i.e. it must be a parallelotope (parallelepiped), a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron. This statement answers questions of \textit{N. Gravin} et al. [Combinatorica 32, No. 6, 629--648 (2012; Zbl 1299.52019)]. As a comparison, it is known that non-parallelohedral fivefold translative tiles exist in every dimension. Also the structures of a belt of any fivefold and sixfold translative tiles in \(\mathbb{E}^3\) are shown.
scientific article; zbMATH DE number 7622200
scientific article