Construction of Crystallographic Tilings by the Cut-and-Project Method

Abstract

Patterns of point sets and tilings were produced and studied for centuries in many cultures. At the border line of mathematics, crystallography, chemistry and physics the general opinion is that every construction for point sets has a counterpart for tilings, but usually this belief is not supported by theorems. This thesis explores the connection between point sets and tilings in two areas in more details: (1) The construction of tilings from point sets, leading to the introduction of clustered Delone point sets and the clustered Voronoi cell tiling construction, which allows to produce tilings from point sets which cannot be constructed as Voronoi cell tilings. (2) A generalized cut-and-project method to produce tilings from higher-dimensional hypercube tilings, including non-generic boundary cases. This allows to construct crystallographic tilings as cut-and-project tilings.