Hexagonal and Trigonal Quasiperiodic Tilings

Abstract

Exploring nonminimal‐rank quasicrystals, which have symmetries that can be found in both periodic and aperiodic crystals, often provides new insight into the physical nature of aperiodic long‐range order in models that are easier to treat. Motivated by the prevalence of experimental systems exhibiting aperiodic long‐range order with hexagonal and trigonal symmetry, we introduce a generic two‐parameter family of 2‐dimensional quasiperiodic tilings with such symmetries. We focus on the special case of trigonal and hexagonal Fibonacci, or golden‐mean, tilings, analogous to the well studied square Fibonacci tiling. We first generate the tilings using a generalized version of de Bruijn's dual grid method. We then discuss their interpretation in terms of projections of a hypercubic lattice from six dimensional superspace. We conclude by concentrating on two of the hexagonal members of the family, and examining a few of their properties more closely, while providing a set of substitution rules for their generation.