Nonperiodicity implies unique composition for self-similar translationally finite tilings.

*(English)*Zbl 0919.52017Let \({\mathcal T}\) 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.

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.

Reviewer: Ch.Leytem (Cruchten)

##### MSC:

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

##### Keywords:

nonperiodic crystallographic; translationally finite tiling; self-similar; unique composition
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\textit{B. Solomyak}, Discrete Comput. Geom. 20, No. 2, 265--279 (1998; Zbl 0919.52017)

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