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Modern crystallography faces a demanding challenge of describing atomic structure and diffraction pattern of quasicrystals, which, after 30 years of Shechtman’s discovery, is still an open field of research. The classical approach based on the Braggs and Laue equations in three-dimensional space is useless, because the direct and the reciprocal lattices cannot be introduced for aperiodic systems. A standard solution to this problem, applied by number of scientists, is to retrieve periodicity in high dimensions. This is a purely mathematical approach with some difficulties from a point of view of physics. It is mathematically elegant, but not applicable to all aperiodic systems (e.g. Thue-Morse or Rudin-Shapiro sequences). It meets also a serious trouble in a proper description of structural defects, like phasons. In our opinion the most successful alternative to the multidimensional description is a statistical method of diffractional and structural analysis of aperiodic systems, also known as the average unit cell approach (AUC). In this work an application of the AUC method to selected aperiodic systems, including modulated structures, quasicrystals and covering clusters, is discussed in the form of a mini-review. A reader can find more details in the cited references.
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