Ever since the discovery of quasicrystals, periodic approximants of these aperiodic structures constitute a very useful experimental and theoretical device. Characterized by packing motifs typical for quasicrystals arranged in large unit cells, these approximants bridge the gap between periodic and aperiodic positional order. Here we propose a class of sequences of 2-D quasicrystals that consist of increasingly larger periodic domains and are marked by an ever more pronounced periodicity, thereby representing aperiodic approximants of a periodic crystal. Consisting of small and large triangles and rectangles, these tilings are based on the metallic means of multiples of 3, have a 6-fold rotational symmetry, and can be viewed as a projection of a non-cubic 4-D superspace lattice. Together with the non-metallic-mean three-tile hexagonal tilings, they provide a comprehensive theoretical framework for the complex structures seen, e.g., in some binary nanoparticles, oxide films, and intermetallic alloys.

自从发现准晶体以来，这些非周期性结构的周期近似值就构成了非常有用的实验和理论装置。这些近似值的特征是堆积在大晶胞中的准晶体的典型堆积图案，从而弥合了周期性和非周期性位置顺序之间的间隙。在这里，我们提出了一类二维准晶体序列，该序列由越来越大的周期性畴组成，并且具有越来越明显的周期性，从而代表了周期性晶体的非周期性近似。这些平铺图由大小的三角形和矩形组成，基于3的倍数的金属平均值，具有6倍的旋转对称性，可以看作是非立方的4-D超空间格的投影。加上非金属均值三块六角形瓷砖，