Aperiodic order and quasicrystals
Aperiodic Order is concerned with the investigation of discrete structures in space, such as point sets or tilings, which possess a certain degree of order while not showing any translational symmetry. A mathematical introduction to the field is provided by the book Aperiodic Order, recently published by Cambridge University Press.
Quasicrystals - aperiodic order in chemistry
Some of the motivation to consider aperiodically ordered structures comes from nature. In 1982, Dan Shechtman discovered crystal-like structures in intermetallic alloys which are well ordered, but show icosahedral symmetry which is incompatible with lattice periodicity. He was awarded the Nobel Prize in Chemistry 2011 for this discovery of what is now known as quasicrystals. From a mathematical point of view, the main aim is to understand not only properties of aperiodically ordered structures, but really to gain a deeper understanding of what types of order exist (and currently there is no mathematical definition of what "order" means, even though everybody has an intuitive understanding of this term) and how we can classify ordered structures.
The so-called pinwheel tiling, on the façade of a building in Melbourne's federation square.
Research at the OU
Uwe Grimm's current research is mainly concerned with the mathematical diffraction of aperiodic structures. This is one measure of order, and corresponds to the diffraction seen in X-ray experiments on aperiodically ordered solids.