In this talk I will discuss a Minkowski decomposition of the associahedron into Bruhat interval polytopes and a mixed subdivision of this into n! cubes. The cubulation has the same combinatorial structure as a known subdivision of the permutahedron and is related to the bracket vector construction of the Tamari lattice. The results of the talk come from a toric geometry perspective which allows us to subdivide many brick polytopes and suggests interesting new generalizations of brick polytopes.