  For a natural class of $r \times n$ integer matrices, we construct a
non-convex polytope which periodically tiles $\mathbb R^n$. From this tiling,
we provide a family of geometrically meaningful maps from a generalized
sandpile group to a set of generalized spanning trees which give multijective
proofs for several higher-dimensional matrix-tree theorems. In particular,
these multijections can be applied to graphs, regular matroids, cell complexes
with a torsion-free spanning forest, and representable arithmetic matroids with
a multiplicity one basis. This generalizes a bijection given by Backman, Baker,
and Yuen and extends work by Duval, Klivans, and Martin.
