Andreas W.M. Dress
The mathematical study of highly regular spatial structures has obvious implications in chemistry, physics and crystallography, relating to crystals as well as to giant molecules like fullerenes and polyoxometalates. Amazingly enough, it has turned out that algebra, and - more specifically - combinatorial group theory (including the theory of orbifolds), does not only play a central role in the description and classification of the symmetries of such structures, but also helps a great deal regarding the analysis of topological aspects.
The starting point of our investigations was the observation that any periodic tiling of any simply connected n-manifold can be represented by a so-called Delaney symbol which is nothing but a (generally finite) set together with an action of a certain finitely presented group (more precisely, the free Coxeter group in n+1 generators) and a labelling of its elements by coxeter matrices satisfying a number of compatibility conditions. This representation is not only uniquely determined (up to isomorphism of symbols) by the tiling, - much more importantly, it can be shown that tilings with isomorphic symbols are necessarily equivariantly homeomorphic. So, we have a canonical embedding of the world of tilings into the algebraically tractable world of symbols. Based on this fundamental fact, theoretical and practical tools have been developed during the past 15 years for the classification, construction and visualization of periodic tilings, especially in the three classical two-dimensional geometries.
As the two-dimensional case is now quite well understood, recent efforts of our group have been concentrating on tilings in euclidean three-space, where a basic issue is to decide whether a given well-formed symbol really corresponds to such a tiling, and, if so, to construct such a tiling. A practical partial solution has been found and implemented on the computer to act as a filter for the large lists of potential solutions produced by enumeration programs. As a consequence, several classification problems have already been solved almost automatically. For instance, Olaf Delgado Friedrichs and Daniel Huson could show that there are exactly 9 (topological types of) tilings of the euclidean 3-space by tetrahedral tiles so that some symmetry group acts transitively on the set of tiles, and they could also derive corresponding results for tilings with cubic, octahedral and other tiles and for tetrahedral tilings with more than one symmetry class (orbit) of tiles. Currently, applications to the enumeration and analysis of certain structure types of crystals are under way which will also be reported.