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
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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