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.