Averaged coordination numbers of planar aperiodic tilings

TL;DR

This paper investigates averaged coordination numbers in planar aperiodic tilings, especially Ammann-Beenker, revealing their structure and relation to topological invariants, with explicit calculations for large distances.

Contribution

It introduces a method to compute averaged coordination numbers in aperiodic tilings, linking them to topological invariants and providing explicit results for Ammann-Beenker tiling.

Findings

Coordination shells form complete shelling orbits in Ammann-Beenker tiling

Explicit calculation of averaged coordination numbers for large distances

Connection between coordination numbers and topological invariants

Abstract

We consider averaged shelling and coordination numbers of aperiodic tilings. Shelling numbers count the vertices on radial shells around a vertex. Coordination numbers, in turn, count the vertices on coordination shells of a vertex, defined via the graph distance given by the tiling. For the Ammann-Beenker tiling, we find that coordination shells consist of complete shelling orbits, which enables us to calculate averaged coordination numbers for rather large distances explicitly. The relation to topological invariants of tilings is briefly discussed.