25 Years of Quasiperiodic Crystallography in Physical Space using the Average Unit Cell Approach

Since the discovery of quasicrystals 40 years ago, many new paradigms and methods have been introduced to crystallography. 25 years ago, a statistical method of structure and diffraction analysis of aperiodic materials was proposed and, over these years, developed to describe model and real systems. This short review paper briefly invokes the basic concepts of the method: a reference lattice and an average unit cell, but also gives an overview of its application to atomic structure and diffraction analysis of various systems. Results are briefly discussed for mathematical sequences (Fibonacci and Thue-Morse), model quasilattices in 2D and 3D (Penrose and Ammann tiling), refinements of real decagonal and icosahedral quasicrystals, analysis of structure disorder in quasicrystals, description of modulated systems, including macromolecular biological systems, and beyond usual application in crystallography.