Canonical-Cell Tilings and their Atomic Decorations

The canonical cell tiling is a geometrical framework that uses four kinds of basic polyhedra, called the canonical cells, to model the packing of atoms and clusters in icosahedral quasicrystals and related periodic approximants. Over the past three decades, it has become increasingly clear that this framework is the most sensible approach to describe related structures, albeit technically much less tractable than the Ammann-Kramer-Neri tiling, which is the simplest icosahedral tiling geometry based on the two Ammann rhombohedra. Geometrical arrangements of cells pose a number of combinatorial problems that cannot be handled using simple linear algebra, making it infeasible to determine structures using the standard six-dimensional scheme. This up-to-date review begins with the motivation, definition, and mathematical facts about the canonical cell tiling. Then the reader is taken through the zoo of concrete structures, from smaller periodic approximants to larger ones, along with an overview of the techniques and heuristics used to study them. The recent discovery of a quasiperiodic canonical cell tiling is also briefly illustrated. The latter half of this review surveys the atomistic modeling of real atomic structures in all the three existing structural families based on the decoration concept of the canonical cell tiling.