Tilings by related zonotopes

We study periodic tilings of d-dimensional space by clusters of related zonotopes. For such tilings, the fundamental region relative to a fully-dimensional translation group is the finite union of zonotopes constructed as Minkowski sums of subsets of a given set of generating vectors. For any rationally-realizable zonotope Z, the rational realization itself determines a unique periodic tiling in which one of the cells is the zonotope Z. In an effort to find analogous tilings using nonrational zonotopes, we conjecture a characterization in terms of a new family oriented matroid structures, for which we provide axioms.