Fractals Generated by Modifying Aperiodic Substitution Tilings

TL;DR

This paper introduces a method to generate an infinite variety of fractals using aperiodic substitution tilings, specifically applying higher-order substitutions to the Ammann Chair tiling and analyzing their dimensions.

Contribution

It presents a novel approach to creating fractals from aperiodic tilings and provides a publicly available fractal generator implementation.

Findings

Fractals generated have dense similarity dimensions.

Higher-order substitutions relate to the Sierpinski carpet.

A fractal generator was successfully implemented in Java.

Abstract

This study proposes a method for producing an infinite number of fractals using aperiodic substitution tilings, exemplified by the Ammann Chair tiling. Higher-order substitutions of aperiodic tilings are utilized in relation to the Sierpinski carpet concept. The similarity dimensions of the fractals generated by the Ammann Chair tiling are calculated and shown to be dense. A fractal image generator was implemented in the Java programming language and is freely available for public use at https://github.com/KahHengLee/Ammann-Chair-Fractal.git