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.
Highlights
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]
This set of tiles, which are forced by a given vertex type, or in this paper extended to any local patch of tiling, represents the empire of the local patch in the quasicrystal [2], a term originally coined by Conway [3]
First we identify the intersections in a sample patch in the pentagrid (a) and we construct a dual quasicrystal cell, here the prolate and oblate rhombuses, at each intersect point, placing them edge to edge while maintaining their topological connectedness (b)
Summary
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]. It is true that a given local patch of tiles in a quasicrystal can force tiles to lie in non-adjacent (non-local) positions. We discuss three methods for generating the empires of the vertex configurations in quasicrystals, using Penrose tiling as an example. We demonstrate how to obtain the empires of a 2D quasicrystal, Penrose tiling, using the empires of this 1D Fibonacci chain This method was first introduced in [2] and named the Fibonacci chain method in this paper.
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