  One well studied way to construct quasicrystalline tilings is via
inflate-and-subdivide (a.k.a. substitution) rules. These produce self-similar
tilings--the Penrose, octagonal, and pinwheel tilings are famous examples. We
present a different model for generating hierarchical tilings we call "fusion
rules". Inflate-and-subdivide rules are a special case of fusion rules, but
general fusion rules are more flexible and allow for defects, changes in
geometry, and even constrained randomness. A condition that produces
homogeneous structures and a method for computing frequency for fusion tiling
spaces are discussed.
