Fusion: a General Framework for Hierarchical Tilings

N.P. Frank Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, U.S.A.

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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.

DOI: 10.12693/APhysPolA.126.461 PACS numbers: 61.44.Br; 02.40.Gh; 61.44.Br