Causal cells: spacetime polytopes with null hyperfaces

We consider polyhedra and 4-polytopes in Minkowski spacetime - in particular, null polyhedra with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces. We present the basic properties of several classes of null-faced 4-polytopes: 4-simplices, "tetrahedral diamonds" and 4-parallelotopes. We propose a "most regular" representative of each class. The most-regular parallelotope is of particular interest: its edges, faces and hyperfaces are all congruent, and it features both null hyperplanes and null segments. A tiling of spacetime with copies of this polytope can be viewed alternatively as a lattice with null edges, such that each point is at the intersection of four lightrays in a tetrahedral pattern. We speculate on the relevance of this construct for discretizations of curved spacetime and for quantum gravity.