2 A. FORREST, J. HUNTON AND J. KELLENDONK share this unusual property and in recent studies they have become the prefered model of material quasicrystals [1] [2]. This model is not without criticism, see e.g. [La2]. Whatever the physical significance of the projection method con- sruction, it also has great mathematical appeal in itself: it is ele- mentary and geometric and, once the acceptance domain and the dimensions of the spaces used in the construction are chosen, has a finite number of degrees of freedom. The projection method is also a natural generalization of low dimensional examples such as Sturmian sequences [HM] which have strong links with classical diophantine approximation. General approach of this book In common with many papers on the topology of tilings, we are motivated by the physical applications and so are interested in the properties of an individual quasicrystal or pattern in Euclidean space. The topological invariants of the title refer not to the topological arrangement of the particular configuration as a subset of Euclidean space, but rather to an algebraic object (graded group, vector space etc.) associated to the pattern, and which in some way captures its geometric properties. It is defined in various equivalent ways as a classical topological invariant applied to a space constructed out of the pattern. There are two choices of space to which to apply the invariant, the one C*-algebraic, the other dynamical, and these reflect the two main approaches to this subject, one starting with the construction of an operator algebra and the other with a topological space with Md action. The first approach, which has the benefit of being closer to phys- ics and which thus provides a clear motivation for the topology, can be summarized as follows. Suppose that the point set T represents the positions of atoms in a material, like a quasicrystal. It then provides a discrete model for the configuration space of particles moving in the material, like electrons or phonons. Observables for these particle systems, like energy, are, in the absence of external forces like a mag- netic field, functions of partial translations. Here a partial translation is an operator on the Hilbert space of square summable functions on T which is a translation operator from one point of T to another com- bined with a range projection which depends only on the neighbouring configuration of that point. The appearance of that range projection is directly related to the locality of interaction. The norm closure AT of the algebra generated by partial translations can be regarded as the (7*-algebra of observables. The topology we are interested in is

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