Fang Fang, Dugan Hammock, and Klee Irwin (2017)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.
crystals
Article
Methods for Calculating Empires in Quasicrystals
Fang Fang *,
ID , Dugan Hammock and Klee Irwin
ID
Quantum Gravity Research, Los Angeles, CA 90290, USA; dugan@QuantumGravityResearch.org (D.H.);
Klee@quantumgravityresearch.org (K.I.)
* Correspondence: fang@quantumgravityresearch.org; Tel.: +1-310-574-6934
Academic Editor: Dmitry A. Shulyatev
Received: 31 August 2017; Accepted: 29 September 2017; Published: 9 October 2017
Abstract: 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.
Keywords: quasicrystals; empires; forced tiles; cut-and-project; defects
1. Introduction—What Is the Empire Problem?
When compared with regular crystals, quasicrystals present more complex structures and
variations and have dynamic patterns that can be non-local due to the non-local nature of the
quasicrystal itself [1]. Although not obvious, it is true that a given local patch of tiles in