A real-space method for describing, constructing and understanding quasicrystals

Quasicrystals (QC) are structures with perfect order but lacking translational symmetry. Consequently, they possess some very peculiar properties, such as self-similarity, and exhibit unique internal structural rearrangements called phason flips. The state-of-the-art in understanding quasicrystals today involves the use of higher-dimensional methods, of which the most important is the project-and-cut method. This method requires facility in mapping to 5-D or higher-dimensional space, which for many researchers poses a considerable obstacle to developing an intuitive understanding of the structural complexity of quasicrystals. Although simpler, real-space approaches to understand quasicrystal structure exist (such as inflation/deflation and covering), these approaches are intrinsically unable to describe phason flips. Here we propose a new quasi-unit cell framework for describing, categorizing, constructing and understanding quasicrystals based on their self-similarity. Our framework utilizes a newly developed concept we call layering, which can explain and predict phason flips based solely on the structure of the quasi-unit cell. We show how the new framework applies to several popular 2-dimensional QC models (Penrose tilings, Ammann-Beenker tiling, etc.).