An *A_\infty bialgebra* is as an algebra over the free matrad H_\infty. There is a family of polytopes KK_{n,m} whose cellular chains realize H_\infty. This talk discusses the following 3-step construction KK_{m,n}: (1) Perform an "(m,n)-subdivision" of the codim 1 cell {1,...,m}|{m+1,...,m+n} of P_{m+n}, the permutahedron, (2) appropriately subdivide the other cells of P_{m+n} and (3) construct an appropriate quotient.