The Mathematics of Long-Range Aperiodic Order

THEOREM: Rotational symmetries of order greater than six, and also five-fold rotational symmetry, are impossible for a periodic pattern in the plane or in three-dimensional space. The discovery of quasicrystals shattered this fundamental law, not by showing it to be logically false but by showing that periodicity was not synonymous with long-range order, if by long-range order we mean whatever order is necessary for a crystal to produce a diffraction pat- tern with sharp bright spots. It suggested that we may not know what long-range order means, nor what a crystal is, nor how symmetry should be defined. Since 1984, solid state science has been under going a veritable K uhnian revolution. -M. SENECHAL, Quasicrystals and Geometry Between total order and total disorder He the vast majority of physical structures and processes that we see around us in the natural world. On the whole our mathematics is well developed for describing the totally ordered or totally disordered worlds. But in reality the two are rarely separated and the mathematical tools required to investigate these in-between states in depth are in their infancy.