Average Degree of Graphs Derived From The Ammann A2 Aperiodic Tiling

Speaker: Prof. Dr. Darren Ong Chung Lee

The Ammann A2 tiling is a simple aperiodically ordered tiling of the plane. We consider the graph derived from this tiling by treating each corner of each tile as a vertex and each side of each tile as an edge. We present a closed-form formula for the average degree of the graph corresponding to this Ammann A2 tiling. More precisely, we derive explicit formulas for the number of vertices and the sum of all vertex degrees in each finite generation of the tiling, which then yield the average degree. Our method is based on the recursive structure of the Ammann A2 substitution tiling and a detailed analysis of how vertices on the intersection line change from one generation to the next. We also show that the average degree converges to a limiting value and support the theoretical result with numerical computations. This provides, to our knowledge, the first explicit average-degree formula for the graph associated with the Ammann A2 tiling.