To understand the structural ordering of aperiodic crystals such as incommensurately modulated phases, composite crystals, quasicrystals and their structurally closely related rational approximants, the higher-dimensional (nD) approach is essential. In a properly selected nD space, aperiodic crystal structures can be described as 3D sections/projections of nD periodic “hypercrystal” structures. Furthermore, “non-crystallographic” symmetries of quasicrystals can become compatible with nD lattice symmetry. For instance, 5-, 8-, 10-, and 12-fold rotational symmetries are proper symmetry operations in 5D and icosahedral point group symmetries in 6D hypercubic lattices. First the so-called “strip-projection” (or “cut-and-project”) method is introduced, because it allows a more intuitive understanding of when and how clusters order quasiperiodically. Then the nD section method is presented, which is the method of choice for nD structure analysis, because it can make use of the reciprocal space information experimentally accessible by diffraction methods.
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