{'fraction_non_alphanumeric': 0.03801652892561983, 'fraction_numerical': 0.0, 'mean_word_length': 5.3125, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Phenomenological theory of icosahedral incommensurate ("quasiperiodic") order in Mn-Al alloys.

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.