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For a natural class of r × n integer matrices, we construct a non-convex polytope which periodically tiles 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 arithmeticarXiv:2007.09501v3 fatcat:uiddlnyvwvftpdozryh2mzlqbq