The Latin square of order n is a familiar combinatorial object. It can be regarded as the Cayley table of a group (which is unique up to group isomorphism) if and only if it satisfies a combinatorial condition called the quadrangle criterion.
Alternatively, we can consider a Latin square to be a set of three partitions of the set of n2 cells: the parts are rows, columns and letters respectively. Any two of these partitions, together with the two trivial partitions, give a Cartesian lattice of dimension two, which is sometimes called a grid.
In this talk I will generalize these ideas to Latin cubes. These have many different definitions. For the type that I will consider, I will show that a simple combinatorial condition forces the relevant partitions to be defined by the diagonal subgroup of a group, which is again unique up to isomorphism.