Abstract

By proposing a family of two-lengthscale quasicrystalline tilings characterized by even-numbered metallic-mean inflation ratios, we extend the recently introduced notion of aperiodic approximants of triangular and honeycomb lattices to square crystals. The proposed family originates in the eightfold Ammann-Beenker and fourfold Harriss quasicrystals and is based on sets of two square tiles and a parallelogram. We elaborate the higher-dimensional representation of the new tilings and we show that at large inflation ratios they tend to the square lattice just like their triangular and hexagonal counterparts.