Higher-dimensional crystallography of N-fold quasiperiodic tilings

Crystallography and periodic average structures (PASs) of two-dimensional (2D) quasiperiodic tilings with N-fold symmetry (N-QPTs with N = 7, 8, 9, 10, 11, 12, 13, 15) were studied using the higher-dimensional approach. By identifying the best (most representative) PASs for each case, it was found that the complexity of the PASs and the degree of average periodicity (DAP) strongly depend on the dimensionality and topology of the hypersurfaces (HSs) carrying the structural information. The distribution of deviations from periodicity is given by the HSs projected upon physical space. The 8-, 10- and 12-QPTs with their 2D HSs have the highest DAP. In the case of the 7-, 9-, 11-, 13- and 15-QPTs, the dimensionality of the HSs is greater than two, and is therefore reduced in the projection upon 2D physical space. This results in a non-homogeneous distribution of deviations from the periodic average lattice, and therefore in a higher complexity of the PASs. Contrary to the 7- and 9-QPTs, which still have representative PASs and DAPs, the 11-, 13- and 15-QPTs have a very low DAP.