This paper is a contribution to the study of rational discrete hyperplanes, i.e., sets of points with integer coordinates lying between two parallel planes. Up to translation and symmetry, they are completely determined by a nonzero normal vector \(\textbf{a}\in {\mathbbm {N}}^d\) . If \(\Vert \textbf{a}\Vert _1 > 2^{d-1}\) , there are two approximations \(\textbf{b},\textbf{c}\in {\mathbbm {N}}^d\) of \(\textbf{a}\) , satisfying \(\textbf{a}=\textbf{b}+\textbf{c}\) , such that the discrete hyperplane plane of normal \(\textbf{a}\) can be partitioned into two disjoint sets having, respectively, the combinatorial structure of discrete hyperplanes of normal \(\textbf{b}\) and \(\textbf{c}\) . The result is based on explicit geometrical mappings described by unimodular \(d\times d\) matrices derived from \(\textbf{a}\) and its approximations. It may have practical interest in discrete geometry for the generation and recognition of discrete hyperplanes as well as for the decomposition of boundaries of discrete sets into planar patches.
