This paper is a contribution to the study of rational discrete planes, i.e., sets of points with integer coordinates lying between two parallel planes. Up to translation and symmetry, they are completely determined by a normal vector a ∈ N 3 . Excepted for a few well-identifed cases, it is shown that there are two approximations b, c ∈ N 3 of a, satisfying a = b + c, such that the discrete plane of normal a can be partitioned into two sets having respectively the combinatorial structure of discrete planes of normal b and c. Christoffel graphs are used to compactly encode the structure of discrete planes. This result may have practical interest in discrete geometry for the analysis of planar features.