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