菱形镶嵌上的距离
Distances on Rhombus Tilings
摘要：
已知欧氏平面的单连通区域的菱形镶嵌形成翻转连通空间（翻转是菱形镶嵌上的基本运算，其将由三个菱形组成的六边形旋转180°）。出于对准晶生长模型的研究，我们在这里有兴趣更好地了解“紧”菱形平铺空间是如何翻转连接的。我们引入一个下界（汉明距离）的最小数量的翻转连接两个平铺（翻转距离），我们调查它是否是尖锐的。答案取决于镶嵌中不同边缘方向的数量n：对于n=3（二聚体镶嵌）或n=4（二次体镶嵌）为正，但对于n=5（十次体镶嵌）或更大的n值可能为负。对于n=3和n=4的情况提供了标准证明，而n=5情况的复杂性导致了计算机辅助证明（其主要结果可以很容易地手工检查）。
英文摘要：
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° 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).
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