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The aim of this review is to describe many approaches to modulated crystals and quasicrystals developed in two decades after the introduction of higher-dimensional crystallography in a unified way. Much attention is focused on higher-dimensional crystallography of quasi-crystals, which is under development. After discussions on symmetries of modulated crystals and methods of their structure analysis, many subjects on the analysis of quasicrystals are discussed, which include methods of generating quasiperiodic tilings, their diffraction patterns, similarity transformations, indexing problems, point density of quasicrystals, phason distortion, relations between quasicrystals and their crystalline approximants, model constructions, Patterson, refinement and maximum-entropy methods for quasicrystals, and superstructures in quasicrystals. In particular, the theories for octagonal, decagonal, dodecagonal and icosahedral quasicrystals are given in detail.
Akio Yamamoto (Mon,) studied this question.