Perhaps the first aperiodic (or quasiperiodic) patterns of general fame were Roger Penrose'  tilings - including the aperiodic tiling of the plane by 'kite' and 'dart' shaped polygonal tiles. In general, we say (roughly) that a pattern in Euclidean space is 'quasiperiodic' if it is aperiodic (i.e., has no translational symmetries) and yet local parts of it, of arbitrary but bounded size, repeat indefinitely. Such patterns (in 3 dimensions) have become of particular interest since the early 1980's as possible models for so-called 'quasicrystals', minerals whose atomic arrangements cannot form a classical crystalline lattice, and yet which still have sufficient local order to display sharp Bragg peaks under X-ray scattering.

The Penrose tilings were originally constructed as a 'substitution' system, a method now giving rise to quite a number of such patterns. A second method is the 'projection' method - this gives rise to many more examples, but it is not clear they necessarily have as many nice geometric properties as the substitution patterns. Some examples, including Penrose's, can be constructed from either method; until recently it was not known whether this was the exception or the norm, but the application of the algebro-topological methods of [FHK2] now shows, among much else, that in fact they are the exception.

The method of showing this is classical in strategy, but novel in application and is worth sketching. To a pattern in Euclidean space it is possible to associate various commutative and non-commutative spaces (C* algebras). From these, 'invariants' can be written down, whether based on K-theory, Cech or group cohomology. It turns out that these are all closely related: the K-theory contains rich, 'high-grade' information, the group cohomology allows for computation and the link between them goes through the Cech cohomology. Work of Anderson and Putnam shows that if the pattern you start out with is a substitution pattern, then the rational version of such invariants will be finite dimensional, but the integral version will as often as not be infinitely generated. In [FHK2] it is shown that the 'generic' projection pattern has infinitely generated rational invariants but, in the rare case when they are finitely generated, the integral versions will be finitely generated too.

There are a variety of unsolved questions in this area, from a complete classification of quasiperiodic patterns to the identification and recognition of geometric properties that might make them suitable models for physical quasicrystals. The latter links with the work of the mathematical physicist Jean Bellissard. He has shown that the K-theory of the C* algebras just mentioned associated to the pattern gives good information about the behaviour of the Schroedinger operator associated to a crystal with atomic arrangement corresponding to the pattern.

For a brief survey, see [FHK1]; this is based on a talk one of us gave in a talk to the ICMP in London in 2000. Full details can be found in [FHK2]

[FHK1] Forrest A.H., Hunton J. and Kellendonk J., Cohomology groups for projection point patterns, proceedings of the XIIIth International Congress of Mathematical Physics, (London, 2000), 333--339, Int. Press, Boston, MA, 2001.

[FHK2] Forrest A.H., Hunton J. and Kellendonk J., Topological invariants for projection method patterns, Memoirs of the American Mathematical Society, number 758. American Mathematical Society, 2002 (120 pp). ISBN 0-8218-2965-3.