Methods for Calculating Empires in Quasicrystals

This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth.