##
**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.

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
PDF
BibTeX
XML
Cite

\textit{B. Solomyak}, Discrete Comput. Geom. 20, No. 2, 265--279 (1998; Zbl 0919.52017)

Full Text:
DOI

### References:

[1] | J. Anderson and I. Putnam, Topological invariants for substitution tilings and their associated C*-algebras, Ergodic Theory Dynamical Systems, to appear. · Zbl 1053.46520 |

[2] | Baake, M.; Schlottmann, M., Geometric aspects of tilings and equivalence concepts, 15-21 (1995), Singapore |

[3] | C. Bandt, Self-similar tilings and patterns described by mappings, in The Mathematics of Aperiodic Order—Proceedings of NATO-Advanced Studies Institute, Waterloo, ON, August 1995, to appear. · Zbl 0891.58018 |

[4] | C. Bandt and G. Gelbrich, Classification of self-affine lattice tilings, J. London Math. Soc.50:2 (1994), 581-593. · Zbl 0820.52012 |

[5] | C. Goodman-Strauss, Matching rules and substitution tilings, Preprint, 1996. · Zbl 0941.52018 |

[6] | B. Grünbaum and G.S. Shephard, Tilings and Patterns, Freeman, New York, 1986. |

[7] | R. Kenyon, The construction of self-similar tilings, Geom. Fund. Anal.6:3 (1996), 471-488. · Zbl 0866.52014 |

[8] | J. C. Lagarias and Y. Wang, Self-affine tiles in Rn, Adv. in Math.121:1 (1996), 21-49. · Zbl 0893.52013 |

[9] | W. F. Lunnon and P. A. B. Pleasants, Quasicrystallographic tilings, J. Math. Pures Appl.66 (1987), 217-263. · Zbl 0626.52017 |

[10] | B. Mossé, Puissances de mots et reconnaisabilité des point fixes d’une substitution, Theoret Comput. Sci.99:2 (1992), 327-334. · Zbl 0763.68049 |

[11] | C. Radin, The pinwheel tiling of the plane, Ann. of Math.139 (1994), 661-702. · Zbl 0808.52022 |

[12] | C. Radin, Space tilings and substitutions, Geom. Dedicata55 (1995), 257-264. · Zbl 0835.52018 |

[13] | C. Radin and M. Wolff, Space tilings and local isomorphism, Geom. Dedicata42 (1992), 355-360. · Zbl 0752.52014 |

[14] | M. Senechal, Quasicrystals and Geometry, Cambridge University Press, Cambridge, 1995. · Zbl 0828.52007 |

[15] | B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynamical Systems, to appear. · Zbl 0884.58062 |

[16] | W. Thurston, Groups, Tilings, and Finite State Automata, AMS Colloquium Lecture Notes, American Mathematical Society, Boulder, 1989. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.