The Structure of Cube Tilings Under Symmetry ConditionsReport as inadecuate

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Discrete and Computational Geometry

, Volume 48, Issue 3, pp 777–782

First Online: 19 July 2012Received: 18 November 2011Revised: 27 April 2012Accepted: 02 July 2012


Let m1,…,md be positive integers, and let G be a subgroup of ℤ such that m1ℤ×⋯×mdℤ⊆G. It is easily seen that if a unit cube tiling 0,1+t,t∈T, of ℝ is invariant under the action of G, then for every t∈T, the number |T∩t+ℤ∩0,m1×⋯×0,md| is divisible by |G|. We give sufficient conditions under which this number is divisible by a multiple of |G|. Moreover, a relation between this result and the Minkowski–Hajós theorem on lattice cube tilings is discussed.

KeywordsCube tiling  Download fulltext PDF

Author: Andrzej P. Kisielewicz - Krzysztof Przesławski


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