On the structure of cube tilings of R3 and R4

A family of translates of the unit cube [0, 1)(d) + T = ([0, 1)(d) + t: t is an element of T]. T subset of R-d, is called a cube tiling of Rd if cubes from this family are pairwise disjoint and boolean OR(t is an element of T) [0, 1)(d) + t = R-d. A non-empty set B = B-1 x ... x B-d subset of R-d is a block if there is a family of pairwise disjoint unit cubes [0, 1)(d) + S, S subset of R-d, such that B = boolean OR(t is an element of S)[0,1)(d) -t and for every t, t' is an element of S there is i is an element of {1, ... , d} such that t(i) - t(i)' is an element of Z \ {0}. A cube tiling of R-d is blockable if there is a finite family of disjoint blocks B, vertical bar B vertical bar > 1, with the property that every cube from the tiling is contained in exactly one block of the family B. We construct a cube tiling T of R-4 which, in contrast to cube tilings of R-3, is not blockable. We give a new proof of the theorem saying that every cube tiling of R-3 is blockable. (C) 2012 Elsevier Inc. All rights reserved.