Coordination shells and coordination numbers of the vertex graph of the Ammann–Beenker tiling

The vertex graph of the Ammann–Beenker tiling is a well-known quasiperiodic graph with an eightfold rotational symmetry. The coordination sequence and coordination shells of this graph are studied. It is proved that there exists a limit growth form for the vertex graph of the Ammann–Beenker tiling. This growth form is an explicitly calculated regular octagon. Moreover, an asymptotic formula for the coordination numbers of the vertex graph of the Ammann–Beenker tiling is also proved.