Annals of Combinatorics 1 (1997) 297-311

Classification of Disordered Tilings

Ludwig Balke

Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany

Received May 7, 1997

AMS Subject Classification: 51M20

Abstract. A tiling T is a disordered realization of a periodic tiling P with symmetry group Γ if we can map the complement of a compact set of T onto the quotient P/Γ in such a way that this map respects the features of the tiling T and P. We show that the global type of a 2-dimensional tiling  T is determined by the periodic tiling P it is a disordered realization of, a conjugacy class of Γ which can be associated to T and a winding number. In some cases, we need in addition some kind of orientation. For higher-dimensional tilings of spaces which are simply connected at infinity, e.g. Rn with n ≥ 3, the associated periodic tiling alone is sufficient.

Keywords: tilings, disordered-materials

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