The question whether quasicrystal shapes should be faceted is studied in a simple model of quasicrystalline order. At T=0, the model is proved to yield a completely faceted equilibrium shape in both two and three dimensions. At T>0, an interface model is derived for a two-dimensional Penrose tiling. By mapping it onto a one-dimensional quasiperiodic Schr\"odinger equation, we show that the roughness exponent varies continuously with T at low T.
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Garg et al. (1987) studied this question.