Abstract
We construct an isomorphism from McMullen's polytope algebra, onto the quotient of the algebra of continuous, piecewise polynomial functions with integral value at 0, by its ideal generated by coordinate functions. This explains the non-trivial grading of the polytope algebra, by the obvious grading of piecewise polynomial functions. In the process of the proof, we make explicit many connections between convex polytopes and piecewise polynomials.
