  A classic problem in matroid theory is to find subspace arrangements,
specifically hyperplane and pseudosphere arrangements, whose intersection
posets are isomorphic to a prescribed geometric lattice. Engstr\"om recently
showed how to construct an infinite family of such subspace arrangements,
indexed by the set of finite regular CW complexes. In this note, we compute the
face numbers of these representations (in terms of the face numbers of the
indexing complexes) and give upper bounds on the total number of faces in these
objects. In particular, we show that, for a fixed rank, the total number of
faces in the Engstr\"om representation corresponding to a codimension one
homotopy sphere arrangement is bounded above by a polynomial in the number of
elements of the matroid with degree one less than its rank.
