Fractals Generated by Modifying Aperiodic Substitution Tilings May 19 2023 Abstract

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Fractals Generated by Modifying Aperiodic Substitution Tilings

May 19, 2023

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.

1 Introduction

A tiling is a plane covering composed of compact sets of tiles that are almost disjoint, in the sense that no

two of them have common interior points. On the other hand, a fractal is a complex geometric shape with

ﬁne structure at arbitrarily small scales, often exhibiting a degree of self-similarity. Fractals are explained

and illustrated in the book Nonlinear Dynamics and Chaos, speciﬁcally in chapter 11 [1]. This paper aims

to connect the concepts of aperiodic tilings and fractals and proposes a method for building fractals from

tilings. The research on tilings contributes to various ﬁelds, such as art, geometry, and even crystallography.

Our study speciﬁcally focuses on the Ammann Chair tiling [2], which exhibits self-similarity properties

that make similarity dimension calculations much simpler. Unlike the well-known aperiodic Penrose tiling,

the Ammann Chair tiling does not lack self-similarity properties. The boundary of the Penrose tiling changes

with each iteration, making it diﬃcult to determine its similarity dimension.

In this paper, we also reference the well-known plane fractal, the Sierpinski Carpet, which was constructed

in 1916 by Waclaw Sierpinski [3]. The fractal’s main idea is to remove a certain area (the middle square) in

each iteration, eventually forming a complex geometric shape with ﬁne holes at arbitrarily small scales after

many iterations.

Figure 1: Sierpinski Carpet and Sierpinski Triangle.

Figure 2: Penrose kite and dart tiling (left), and Ammann Chair tiling (right).

arXiv:2210.15465v2 [math.MG] 18 May 2023

2 Similar Research

Several studies have explored the relationship between fractals and aperiodic tiling, with a particular focus

on the Penrose tiling.

Liu’s research in physics [4] investigates the scaling and scaling-related dynamical properties of the

Penrose tiling. Analytically, the Penrose tiling has a fractal dimension of df= 2, which is equivalent to its

Euclidean dimension (in R2). Additionally, the paper provides numerical evidence that the physics equality

holds on the Penrose lattice. This equality describes the relationship between spectral dimension, fractal

dimension, and diﬀusive dimension. Liu’s research not only provides insights into the physical properties of

the Penrose lattice but also analytical calculations that can be used to determine the fractal dimension of

the Ammann Chair tiling.

In the publication Fractal Dual Substitution Tilings [5], the authors demonstrate a method for creating

an inﬁnite number of new fractal tilings. They achieve this by creating new tilings through graph-iterating

function systems, and the resulting tiles have fractal boundaries. Like our paper, this article shows how to

create an inﬁnite number of new fractals, but with a focus on modifying the tile boundary rather than plane

fractal.

Another article discusses a modiﬁcation of the Penrose tiling to achieve self-similar properties. This

involves changing the shape of the kite and dart (Penrose tiling tiles) from quadrilateral to spiky fractals.

The modiﬁcation includes self-similar properties as well as perfect matching rules [6]. This research provides

an option for extending our paper, namely how to replace the Ammann Chair tiling with the Penrose tiling

while preserving self-similarity.

3 Ammann Chair Tiling

We can generate the Ammann Chair tiling using an algorithm that replaces existing tiles (old) with new

tiles. This method of generating tiling is known as the substitution method.

Figure 3: Tiles of the Ammann Chair tiling.

3.1 The Typical Substitution Method

The Ammann Chair tiling is typically constructed using two sizes of tiles, which we refer to as the “big tile”

(orange) and the “small tile” (yellow). The size of the big tile is larger than the size of the small tile by a

factor of φ1

2, which is determined during the substitution process.

The area of the Ammann Chair tiling increases over generations with the inﬂation factor of φ1

2, eventually

forming a tiling of the plane [2]. However, for technical reasons, we consider a tiling of the initial tile. By

scaling down the size of the tiles over generations, the area of the tiling stays the same. These two approaches

are closely related, except for the diﬀerence in size. Both of them have the same combinatorial and geometric

structures.

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摘要：

FractalsGeneratedbyModifyingAperiodicSubstitutionTilingsMay19,2023AbstractThisstudyproposesamethodforproducinganinnitenumberoffractalsusingaperiodicsubstitutiontilings,exempliedbytheAmmannChairtiling.HigherordersubstitutionsofaperiodictilingsareutilizedinrelationtotheSierpinskicarpetconcept.Thesim...

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Fractals Generated by Modifying Aperiodic Substitution Tilings May 19 2023 Abstract

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