  We prove that fairly general spaces of tilings of R^d are fiber bundles over
the torus T^d, with totally disconnected fiber. This was conjectured (in a
weaker form) in [W3], and proved in certain cases. In fact, we show that each
such space is homeomorphic to the d-fold suspension of a Z^d subshift (or
equivalently, a tiling space whose tiles are marked unit d-cubes). The only
restrictions on our tiling spaces are that 1) the tiles are assumed to be
polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to
full-face), 2) only a finite number of tile types are allowed, and 3) each tile
type appears in only a finite number of orientations. The proof is
constructive, and we illustrate it by constructing a `square' version of the
Penrose tiling system.
