Distances on Rhombus Tilings

TL;DR

This paper investigates the flip-connectedness of rhombus tilings in the plane, introducing a lower bound on flip-distance and analyzing its sharpness across different tiling types, with results supported by theoretical and computational proofs.

Contribution

It introduces a Hamming-distance lower bound for flip-distance in rhombus tilings and studies its sharpness across various edge direction counts, including computational verification for complex cases.

Findings

The Hamming-distance is a sharp lower bound for n=3 and n=4 tilings.

For n=5 or more, the bound may not be sharp, with computational evidence provided.

The study enhances understanding of the connectivity and complexity of rhombus tiling spaces.

Abstract

The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi). Motivated by the study of a quasicrystal growth model, we are here interested in better understanding how "tight" rhombus tiling spaces are flip-connected. We introduce a lower bound (Hamming-distance) on the minimal number of flips to link two tilings (flip-distance), and we investigate whether it is sharp. The answer depends on the number n of different edge directions in the tiling: positive for n=3 (dimer tilings) or n=4 (octogonal tilings), but possibly negative for n=5 (decagonal tilings) or greater values of n. A standard proof is provided for the n=3 and n=4 cases, while the complexity of the n=5 case led to a computer-assisted proof (whose main result can however be easily checked by hand).