New group theoretical methods for applications in virology and quasicrystals

Non-crystallographic symmetries are ubiquitous in physics, chemistry and biology. Prominent examples are quasicrystals, alloys with long-range order but no translational periodicity in their atomic organisation, and viruses, micro-organisms consisting of a protein shell, the capsid, that in most cases displays icosahedral symmetry. Group theory plays a crucial role in understanding their structures and their physical and geometrical properties. In this thesis new group theoretical methods are introduced, to characterise virus organisation and model structural transitions of icosahedral quasicrystals. It is shown that these problems can be described via the embedding of non-crystallographic groups into the point group of higher dimensional lattices. Indeed, the analysis of orbits of such embeddings, akin to the construction of quasicrystals via the cut-and-project method, provides a rigorous mathematical construction of finite nested point sets with non-crystallographic symmetry at each distinct radial level. In the case of icosahedral symmetry, it is shown that the point arrays thus obtained can be used to provide constraints on the geometry of viral capsids, encoding information on the organisation of the capsid proteins and the genomic material collectively. Moreover, structural transitions of icosahedral quasicrystals can be analysed in a group theoretical framework through continuous rotations in the higher dimensional space connecting distinct copies of the embedded icosahedral group, sharing a common maximal subgroup. These rotations induce in projection continuous transformations between aperiodic point sets that preserve the symmetry described by the common subgroup. Theoretical methods as well as applications are discussed, with emphasis on the computational and geometric aspect of group theory.