A three tile 6-fold golden-mean tiling

TL;DR

This paper introduces a novel aperiodic tiling with 6-fold rotational symmetry using three tile types with lengths proportional to 1 and the golden mean, employing substitution rules and analyzing its structure in higher dimensions.

Contribution

It presents a new multi-edge-length aperiodic tiling with 6-fold symmetry, including substitution rules and bipartite vertex analysis, expanding understanding of golden-mean tilings.

Findings

Tiling exhibits 6-fold rotational symmetry.

Vertices form a bipartite structure with a sublattice imbalance.

Analysis extends to 4-dimensional hyperspace.

Abstract

We present a multi-edge-length aperiodic tiling which exhibits 6--fold rotational symmetry. The edge lengths of the tiling are proportional to 1:$\tau$, where $\tau$ is the golden mean $\frac{1+\sqrt{5}}{2}$. We show how the tiling can be generated using simple substitution rules for its three constituent tiles, which we then use to demonstrate the bipartite nature of the tiling vertices. As such, we show that there is a relatively large sublattice imbalance of $1/[2\tau^2]$. Similarly, we define allowed vertex configurations before analysing the tiling structure in 4-dimensional hyperspace.