The periodic average structure of particular quasicrystals

The non-crystallographic symmetry of d-dimensional (dD) quasiperiodic structures is incompatible with lattice periodicity in dD physical space. However, dD quasiperiodic structures can be described as irrational sections of nD (n > d) periodic hypercrystal structures. By appropriate oblique projection of particular hypercrystal structures onto physical space, discrete periodic average structures can be obtained. The boundaries of the projected atomic surfaces give the maximum distance of each atom in a quasiperiodic structure from the vertices of the reference lattice of its average structure. These maximum distances turn out to be smaller than even the shortest atomic bond lengths. The metrics of the average structure of a 3D Ammann tiling, for instance, with edge lengths of the unit tiles equal to the bond lengths in elemental aluminium, correspond almost exactly to the metrics of face-centred-cubic aluminium. This is remarkable since most stable quasicrystals contain aluminium as the main constitutent. The study of the average structure of quasicrystals can be a valuable aid to the elucidation of the geometry of quasicrystal-to-crystal transformations. It can also contribute to the derivation of the physically most relevant Brillouin (Jones) zone.