Self-Similar Transformation and Vertex Configurations of the Octagonal Ammann–Beenker Tiling

Based on the matching rules for squares and rhombuses, we study the self-similar transformation and the vertex configurations of the Ammann–Beenker tiling. The structural properties of the configurations and their relations during the self-similar transformation are obtained. Our results reveal the distribution correlations of the configurations, which provide an intuitive understanding of the octagonal quasi-periodic structure and also give implications for growing perfect quasi-periodic tiling according to the local rules.