Distances on rhombus tilings |
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Authors: | Olivier Bodini,Thomas FerniqueMichael Rao,É ric Ré mila |
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Affiliation: | a LIP6, CNRS & University of Paris 6, 4 place Jussieu 75005 Paris, Franceb Poncelet Lab., CNRS & Independent University of Moscow, 119002, Bolshoy Vlasyevskiy Per. 11, Moscow, Russiac LIP, CNRS & ENS de Lyon & Univ. Lyon 1, 46 allée d’Italie 69007 Lyon, France |
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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° 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 (octagonal 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 manually). |
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Keywords: | Computer-assisted proof Flip Phason Pseudoline arrangement Quasicrystal Rhombus tiling Tiling space |
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