Multiple tilings of Euclidean space by unit cubes
Generalizing a famous problem on (simple) tilings (Keller conjecture, 1930), the author considers multiple tilings by congruent cubes and describes two explicit (counter)examples which avoid cubes that meet face-to-face: a double and a triple (non-lattice) tiling of \( {\mathbb{R}}^3 \). These examples have been found with the help of a group-theoretic characterization which generalizes a result by Hajós (1942) (used in his proof of Minkowski's conjecture, 1896, i.e., the Keller conjecture for lattice tilings) which is stated without proof. It follows that, for all \( k \geq 2 \) and all \( n \geq 3 \), there are \(k\)-fold (non-lattice) tilings of \( {\mathbb{R}}^n \) by congruent cubes such that no two cubes share a common face. (This settles the cases left open by R. M.~Robinson's result (1979) on lattice tilings. For \( n \leq 2 \), each multiple tiling is the union of simple tilings, and thus face-to-face `cubes' cannot be avoided.) Remark: The Keller conjecture that, for \( k=1 \), there always are cubes sharing an \((n-1)\)-dimensional face, known to be true for \( n \leq 6 \) (Perron, 1940), here erroneously described as open, has recently been disproved for \( n \geq 10 \) [\textit{J. C. Lagarias} and \textit{P. W. Shor}, Bull. Am. Math. Soc. 27, No. 2, 279-283 (1992; Zbl 0759.52013)] and thus is unsettled only for \( n=7,8,9\).
scientific article; zbMATH DE number 1471535