  In this paper we show that the pipe dream complex associated to the
permutation 1n(n-1)...2 can be geometrically realized as a triangulation of the
vertex figure of a root polytope. Leading up to this result we show that the
Grothendieck polynomial specializes to the h-polynomial of the corresponding
pipe dream complex, which in certain cases equals the h-polynomial of canonical
triangulations of root (and flow) polytopes, which in turn equals a
specialization of the reduced form of a monomial in the subdivision algebra of
root (and flow) polytopes. Thus, we connect Grothendieck polynomials to reduced
forms in subdivision algebras and root (and flow) polytopes. We also show that
root polytopes can be seen as projections of flow polytopes, explaining that
these families of polytopes possess the same subdivision algebra.
