The construction of the fiber polytope ∑(P, Q) of a projection π:P→Q of polytopes is extended to flags of projections. While the faces of the fiber polytope are related to subdivisions of Q induced by the faces of P, those of an iterated fiber polytope are related to discrete homotopies between polyhedral subdivisions. In particular, in the case of projections
starting with an (n + 1)-simplex, vertices of the successive iterates correspond to, respectively, subsets, permutations and sequences of permutations of an n-set. The first iterate will always be combinatorially an n-cube, and, under certain conditions, the second will have the structure of the (n−1)-dimensional permutohedron.