  We provide a proof of the Alpern multi-tower theorem for Z^d actions. We
reformulate the theorem as a problem of measurably tiling orbits of a Z^d
action by a collection of rectangles whose corresponding sides have no
non-trivial common divisors. We associate to such a collection of rectangles a
special family of generalized domino tilings. We then identify an intrinsic
dynamic property of these tilings, viewed as symbolic dynamical systems, which
allows for a multi-tower decomposition.
