Rational tilings by 𝑛-dimensional crosses

Abstract

Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors e 1 , … , e n {{\mathbf {e}}_1}, \ldots ,{{\mathbf {e}}_n} and whose centers are i e j i{{\mathbf {e}}_j} , 0 ⩽ | i | ⩽ k 0 \leqslant |i| \leqslant k , in n n -dimensional Euclidean space. The union of these cubes is called a cross. This cross consists of 2 k n + 1 2kn + 1 cubes; a central cube together with 2 n 2n arms of length k k . A family of translates of a cross whose union is n n -dimensional Euclidean space and whose interiors are disjoint is a tiling. Denote the set of translation vectors by L {\mathbf {L}} . If the vector set L {\mathbf {L}} is a vector lattice, then we say that the tiling is a lattice tiling. If every vector of L {\mathbf {L}} has rational coordinates, then we say that the tiling is a rational tiling, and, similarly, if every vector of L {\mathbf {L}} has integer coordinates, then we say that the tiling is an integer tiling. Is there a noninteger tiling by crosses? In this paper we shall prove that if there is an integer lattice tiling by crosses, if 2 k n + 1 2kn + 1 is not a prime, and if p > k p > k for every prime divisor p p of 2 k n + 1 2kn + 1 , then there is a rational noninteger lattice tiling by crosses and there is an integer nonlattice tiling by crosses. We will illustrate this in the case of a cross with arms of length 2 in 55 55 -dimensional Euclidean space. Throughout, the techniques are algebraic.