Possible modulated structures of icosahedral quasicrystals are investigated. The modulated phases are regarded as lower symmetric phases where the global icosahedral symmetry of quasicrystals is broken due to mode locking of phason degrees of freedom. With the help of group theory, nonvanishing phason fields are calculated for all the possible symmetry breaking. A modified projection scheme in the presence of nonvanishing phason fields is obtained for modulated quasicrystalline structures with tetrahedral, pentagonal, and trigonal symmetries. Three possible ways of generating a rational approximation of the icosahedral quasicrystal are discussed in connection with the phason degrees of freedom.
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Yasushi Ishii (1989) studied this question.
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