Distances on Lozenge Tilings
In this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the the number of flips – a local transformation on tilings – necessary to link two tilings. We prove that the flip-distance between two tilings is equal to the Hamming-distance for n ≤ 4. We also show, by providing a pair of so-called deficient tilings, that this does not hold for n ≥ 6. We finally discuss the n = 5 case, which remains open.