  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.
