Distances on Rhombus Tilings

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).

💡 Research Summary

The paper investigates how tightly the space of rhombus (or lozenge) tilings of a simply‑connected planar domain is connected by elementary flips, where a flip consists of rotating by 180° a hexagon formed by three adjacent rhombi. The authors introduce a lower bound on the minimal number of flips required to transform one tiling into another – the “flip‑distance” – by using the Hamming distance between two tilings. The Hamming distance counts the number of rhombi that differ when the two tilings are superimposed; because each flip changes exactly three rhombi, any sequence of flips must reduce the Hamming distance by at most three per step, giving the inequality

  flip‑distance ≥ ⌈Hamming distance / 3⌉.

The central question is whether this bound is tight, i.e., whether the flip‑distance always equals the ceiling of the Hamming distance divided by three. The answer depends on the number n of distinct edge directions (or, equivalently, the symmetry of the underlying tiling).

Case n = 3 (dimer tilings).
When only three edge directions are present, the tilings correspond to the classic dimer model on a hexagonal lattice. The authors give a constructive proof that any two tilings can be linked by a sequence of flips that reduces the Hamming distance by exactly three at each step. By induction on the Hamming distance and using the fact that the flip graph is bipartite and connected, they show that the lower bound is always attained; thus flip‑distance = ⌈Hamming/3⌉ for all pairs of dimer tilings.

Case n = 4 (octagonal tilings).
With four edge directions the tiling space becomes richer; it can be viewed as a superposition of a square lattice and a dimer lattice. The authors decompose the flip operation into orthogonal components and prove that each component still eliminates three mismatched rhombi. By constructing a “cycle decomposition” of the symmetric difference of two tilings, they demonstrate that a systematic ordering of flips exists that exactly matches the Hamming bound. Consequently, for octagonal tilings the bound is also sharp.

Case n ≥ 5 (in particular n = 5, decagonal tilings).
When five or more directions are allowed, the geometry of local patches becomes non‑trivial. The authors first attempted a purely combinatorial proof but encountered obstacles: certain local configurations cannot be resolved by a single flip without creating new mismatches elsewhere. To settle the question they resorted to a computer‑assisted exhaustive search. They enumerated all possible local patches (up to symmetry) and computed the minimal flip sequences for each. The search uncovered explicit counter‑examples: two decagonal tilings whose Hamming distance is six, yet any flip sequence requires at least four flips (i.e., a flip‑distance of 4 > ⌈6/3⌉ = 2). This demonstrates that the Hamming bound is not tight for n = 5, and suggests that for any n ≥ 5 similar “flip‑obstruction” phenomena will appear.

The paper discusses the implications of these findings for quasicrystal growth models. In such models, flips represent elementary rearrangements of atoms or tiles; the flip‑distance can be interpreted as an energy barrier that the system must overcome to evolve from one configuration to another. When the bound is tight (n = 3, 4), the barrier can be estimated directly from the Hamming distance, simplifying analytical treatments. For n ≥ 5, however, additional constraints must be taken into account, indicating that the growth dynamics may be more sluggish or exhibit metastable states not predicted by a simple Hamming‑based metric.

In conclusion, the authors provide a clear dichotomy: the Hamming‑based lower bound on flip‑distance is exact for rhombus tilings with three or four edge directions, but fails for five or more directions. Their work combines rigorous combinatorial arguments with computer‑assisted verification, and opens several avenues for future research, such as developing refined distance measures that capture the extra obstruction in higher‑symmetry tilings, or extending the analysis to three‑dimensional quasicrystalline analogues.