  This paper presents an algebraic construction of Euler-Maclaurin formulas for
polytopes. The formulas obtained generalize and unite the previous lattice
point formulas of Morelli and Pommersheim-Thomas, and the Euler-Maclaurin
formulas of Berline-Vergne While the approach of this paper originates in the
theory of toric varieties, and recovers previous results about characteristic
classes of toric varieties, the present paper is self-contained and does not
rely on results from toric geometry. We aim in particular to exhibit in a
combinatorial way ingredients such as such Todd classes and cycle-level
intersections in Chow rings, that first entered the theory of polytopes from
algebraic geometry.
