  Brenti and Welker have shown that for any simplicial complex X, the face
vectors of successive barycentric subdivisions of X have roots which converge
to fixed values depending only on the dimension of X. We improve and generalize
this result here. We begin with an alternative proof based on geometric
intuition. We then prove an interesting symmetry of these roots about the real
number -2. This symmetry can be seen via a nice algebraic realization of
barycentric subdivision as a simple map on formal power series in two
variables. Finally, we use this algebraic machinery with some geometric
motivation to generalize the combinatorial statements to arbitrary subdivision
methods: any subdivision method will exhibit similar limit behavior and
symmetry. Our techniques allow us to compute explicit formulas for the values
of the limit roots in the case of barycentric subdivision.
