I will discuss how one can organize tautological intersection numbers on the moduli space of stable curves into families of lattice point counts of integer dilates of partial polytopal complexes. Viewing these intersection numbers as lattice point counts provides a novel enumerative interpretation. In order to establish this correspondence between intersection numbers and lattice point counts, we first organize tautological intersection numbers into polynomial families. One then uses a classification theorem of Breuer that says these polynomials are actually Ehrhart polynomials. The talk will assume no background in Ehrhart theory. I will end the talk with some open problems, and outline some directions for future work.