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Nonperiodicity implies unique composition for self-similar translationally finite tilings. (English) Zbl 0919.52017
Let \({\mathcal T}\) be a translationally finite tiling of \(\mathbb{R}\) (a tiling which, up to translation, has finitely many patches of diameter less than a given number), which is self-similar (the tiles can be grouped into patches to form a new tiling that is isomorphic to the original one). The tiling \({\mathcal T}\) is said to have the unique composition property, if there is only one way to group its tiles to form a self-similar tiling.
The author proves that if such a tiling \({\mathcal T}\) is nonperiodic (i.e. there is no nonzero vector \(v\) such that \({\mathcal T}+v={\mathcal T}\)), then it has a unique composition property. More generally he proves that \({\mathcal T}\) has the unique composition property modulo its translation symmetries. Thus the tilings obtained as the result of grouping the tiles of \({\mathcal T}\) into a self-similar tiling in two different ways, differ by a translation.

MSC:
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
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