Hexagonal and trigonal quasiperiodic tilings

TL;DR

This paper introduces a new family of 2D quasiperiodic tilings with hexagonal and trigonal symmetries, generated via a generalized dual grid method and hypercubic lattice projections, relevant to understanding aperiodic order in physical systems.

Contribution

It presents a novel two-parameter family of quasiperiodic tilings with specific symmetries, including generation methods and detailed analysis of particular cases.

Findings

Generated tilings using a generalized de Bruijn's dual grid method

Interpreted tilings as projections from six-dimensional superspace

Provided substitution rules for tiling generation

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.