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The crystal structure with icosahedral point-group symmetry and long-range orientational order observed by Shechtman et al. in Mn-Al alloys can be explained as a multiple-q density-wave structure with only one length scale. Its existence and stability can be understood from a simple Landau theory. The diffraction spots can be indexed by six- (or more-) dimensional space-group symmetries. Icosahedral incommensurate structures constitute natural extensions of smectic, rodlike, and cubic crystal structures.
Per Bak (Mon,) studied this question.