Some examples of quasiperiodic tilings obtained with a simple grid method

TL;DR

This paper introduces a simple grid method for creating quasiperiodic tilings with various symmetries, extending previous work by demonstrating new examples using fundamental domains of different lattices.

Contribution

It presents a straightforward grid-based approach to generate quasiperiodic tilings, expanding on prior methods by including new examples with different lattice and fundamental domain configurations.

Findings

Generated quasiperiodic tilings with twelve-fold symmetry.

Extended grid method to new lattice and fundamental domain configurations.

Demonstrated versatility of the grid approach for different tiling patterns.

Abstract

A grid method using tiling by fundamental domain of simple 2D lattices is presented. It refer to a previous work done by Stampfli in $1986$ using two tilings by regular hexagons, one rotate by $\pi/2$ relatively to the other. This allows to get a quasiperiodic structure with a twelve fold symmetry. The quasiperiodic structure is a tiling of the plane by regular triangles, squares and rhombuses. This can be extented to other examples of tilings by fundamental domain. Two other examples are proposed. The first example also based on the hexagonal lattice, but with grids defined by the fundamental rhombic domain formed by two regular triangles. The second example presents the case of a square lattice with a square fundamental domain.