Triangular and dice quasicrystals modulated by generic 1D aperiodic sequences

TL;DR

This paper introduces a method to generate hexagonal aperiodic tilings, including triangular and dice lattices, using 1D aperiodic sequences, and analyzes their diffraction patterns to reveal properties of the underlying sequences.

Contribution

It provides a novel framework for constructing higher-dimensional aperiodic systems based on 1D sequences, extending aperiodic-crystal research beyond one dimension.

Findings

Diffraction patterns show pure point or singular continuous spectra.

Examples include tilings based on Fibonacci, Thue-Morse, and tribonacci sequences.

The method enables systematic creation of aperiodic hexagonal lattices.

Abstract

We present a method for generating hexagonal aperiodic tilings that are topologically equivalent to the triangular and dice lattices. This approach incorporates aperiodic sequences into the spacing between three sets of grids for the triangular lattice, resulting in "modulated triangular lattices". Subsequently, by replacing the triangles with rhombuses, parallelograms, or hexagons, modulated dice or honeycomb lattices are constructed. Using generalized Fibonacci, Thue-Morse, and tribonacci sequences, we demonstrate several examples of hexagonal aperiodic tilings. Structural analysis confirms that their diffraction patterns reflect the properties of the one-dimensional aperiodic sequences, namely pure point (Bragg peaks) or singular continuous. Our method establishes a general framework for constructing a broad range of hexagonal aperiodic systems, advancing aperiodic-crystal research into higher dimensions that were previously focused on one-dimensional aperiodic sequences.