A Quasiperiodic Tiling With 12-Fold Rotational Symmetry and Inflation Factor 1 + Sqrt(3)

TL;DR

This paper introduces a new quasiperiodic tiling with 12-fold rotational symmetry, derived from substitution rules involving specific base tiles, and demonstrates how to generate arbitrarily large symmetric patterns.

Contribution

It presents novel substitution rules for a quasiperiodic tiling with 12-fold symmetry and an inflation factor of 1 + sqrt(3), including variations and an interactive exploration tool.

Findings

The tiling exhibits quasiperiodic 12-fold rosettes.

Arbitrarily large symmetric tilings can be generated.

The tiling uses a square, a rhomb, and half-triangles with specific substitution rules.

Abstract

We show how we found substitution rules for a quasiperiodic tiling with local rotational symmetry and inflation factor 1 + sqrt(3). The base tiles are a square, a rhomb with an acute angle of 30 degrees, and equilateral triangles that are cut in half. These half-triangles follow three different substitution rules and can be recombined into equilateral triangles in nine different ways to make minor variations of the tiling. The tiling contains quasiperiodically repeated 12-fold rosettes. A central rosette can be enlarged to make an arbitrarily large tiling with 12-fold rotational symmetry. An online computer program is provided that allows the user to explore the tiling.