FRACTAL

GEOMETRY

Mathematical Foundations

and Applications

FractalGeometry:MathematicalFoundationsandApplication. Second Edition Kenneth Falconer

2003 John Wiley & Sons, Ltd ISBNs: 0-470-84861-8 (HB); 0-470-84862-6 (PB)

FRACTAL

GEOMETRY

Mathematical Foundations

and Applications

Second Edition

Kenneth Falconer

University of St Andrews, UK

Copyright 2003 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,

West Sussex PO19 8SQ, England

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Contents

Preface ix

Preface to the second edition xiii

Course suggestions xv

Introduction xvii

Notes and references xxvii

PART I FOUNDATIONS 1

Chapter 1 Mathematical background 3

1.1 Basic set theory 3

1.2 Functions and limits 6

1.3 Measures and mass distributions 11

1.4 Notes on probability theory 17

1.5 Notes and references 24

Exercises 25

Chapter 2 Hausdorff measure and dimension 27

2.1 Hausdorff measure 27

2.2 Hausdorff dimension 31

2.3 Calculation of Hausdorff dimension—simple examples 34

*2.4 Equivalent deﬁnitions of Hausdorff dimension 35

*2.5 Finer deﬁnitions of dimension 36

2.6 Notes and references 37

Exercises 37

Chapter 3 Alternative deﬁnitions of dimension 39

3.1 Box-counting dimensions 41

3.2 Properties and problems of box-counting dimension 47

v

vi Contents

*3.3 Modiﬁed box-counting dimensions 49

*3.4 Packing measures and dimensions 50

3.5 Some other deﬁnitions of dimension 53

3.6 Notes and references 57

Exercises 57

Chapter 4 Techniques for calculating dimensions 59

4.1 Basic methods 59

4.2 Subsets of ﬁnite measure 68

4.3 Potential theoretic methods 70

*4.4 Fourier transform methods 73

4.5 Notes and references 74

Exercises 74

Chapter 5 Local structure of fractals 76

5.1 Densities 76

5.2 Structure of 1-sets 80

5.3 Tangents to

s

-sets 84

5.4 Notes and references 89

Exercises 89

Chapter 6 Projections of fractals 90

6.1 Projections of arbitrary sets 90

6.2 Projections of

s

-sets of integral dimension 93

6.3 Projections of arbitrary sets of integral dimension 95

6.4 Notes and references 97

Exercises 97

Chapter 7 Products of fractals 99

7.1 Product formulae 99

7.2 Notes and references 107

Exercises 107

Chapter 8 Intersections of fractals 109

8.1 Intersection formulae for fractals 110

*8.2 Sets with large intersection 113

8.3 Notes and references 118

Exercises 119

PART II APPLICATIONS AND EXAMPLES 121

Chapter 9 Iterated function systems—self-similar and self-afﬁne sets 123

9.1 Iterated function systems 123

9.2 Dimensions of self-similar sets 128

vii

9.3 Some variations 135

9.4 Self-afﬁne sets 139

9.5 Applications to encoding images 145

9.6 Notes and references 148

Exercises 149

Chapter 10 Examples from number theory 151

10.1 Distribution of digits of numbers 151

10.2 Continued fractions 153

10.3 Diophantine approximation 154

10.4 Notes and references 158

Exercises 158

Chapter 11 Graphs of functions 160

11.1 Dimensions of graphs 160

*11.2 Autocorrelation of fractal functions 169

11.3 Notes and references 173

Exercises 173

Chapter 12 Examples from pure mathematics 176

12.1 Duality and the Kakeya problem 176

12.2 Vitushkin’s conjecture 179

12.3 Convex functions 181

12.4 Groups and rings of fractional dimension 182

12.5 Notes and references 184

Exercises 185

Chapter 13 Dynamical systems 186

13.1 Repellers and iterated function systems 187

13.2 The logistic map 189

13.3 Stretching and folding transformations 193

13.4 The solenoid 198

13.5 Continuous dynamical systems 201

*13.6 Small divisor theory 205

*13.7 Liapounov exponents and entropies 208

13.8 Notes and references 211

Exercises 212

Chapter 14 Iteration of complex functions—Julia sets 215

14.1 General theory of Julia sets 215

14.2 Quadratic functions— the Mandelbrot set 223

14.3 Julia sets of quadratic functions 227

14.4 Characterization of quasi-circles by dimension 235

14.5 Newton’s method for solving polynomial equations 237

14.6 Notes and references 241

Exercises 242

viii Contents

Chapter 15 Random fractals 244

15.1 A random Cantor set 246

15.2 Fractal percolation 251

15.3 Notes and references 255

Exercises 256

Chapter 16 Brownian motion and Brownian surfaces 258

16.1 Brownian motion 258

16.2 Fractional Brownian motion 267

16.3 L

´

evy stable processes 271

16.4 Fractional Brownian surfaces 273

16.5 Notes and references 275

Exercises 276

Chapter 17 Multifractal measures 277

17.1 Coarse multifractal analysis 278

17.2 Fine multifractal analysis 283

17.3 Self-similar multifractals 286

17.4 Notes and references 296

Exercises 296

Chapter 18 Physical applications 298

18.1 Fractal growth 300

18.2 Singularities of electrostatic and gravitational potentials 306

18.3 Fluid dynamics and turbulence 307

18.4 Fractal antennas 309

18.5 Fractals in ﬁnance 311

18.6 Notes and references 315

Exercises 316

References 317

Index

329

Preface

I am frequently asked questions such as ‘What are fractals?’, ‘What is fractal

dimension?’, ‘How can one ﬁnd the dimension of a fractal and what does it

tell us anyway?’ or ‘How can mathematics be applied to fractals?’ This book

endeavours to answer some of these questions.

The main aim of the book is to provide a treatment of the mathematics asso-

ciated with fractals and dimensions at a level which is reasonably accessible to

those who encounter fractals in mathematics or science. Although basically a

mathematics book, it attempts to provide an intuitive as well as a mathematical

insight into the subject.

The book falls naturally into two parts. Part I is concerned with the general

theory of fractals and their geometry. Firstly, various notions of dimension and

methods for their calculation are introduced. Then geometrical properties of frac-

tals are investigated in much the same way as one might study the geometry of

classical ﬁgures such as circles or ellipses: locally a circle may be approximated

by a line segment, the projection or ‘shadow’ of a circle is generally an ellipse,

a circle typically intersects a straight line segment in two points (if at all), and

so on. There are fractal analogues of such properties, usually with dimension

playing a key r

ˆ

ole. Thus we consider, for example, the local form of fractals,

and projections and intersections of fractals.

Part II of the book contains examples of fractals, to which the theory of the

ﬁrst part may be applied, drawn from a wide variety of areas of mathematics

and physics. Topics include self-similar and self-afﬁne sets, graphs of functions,

examples from number theory and pure mathematics, dynamical systems, Julia

sets, random fractals and some physical applications.

There are many diagrams in the text and frequent illustrative examples. Com-

puter drawings of a variety of fractals are included, and it is hoped that enough

information is provided to enable readers with a knowledge of programming to

produce further drawings for themselves.

It is hoped that the book will be a useful reference for researchers, providing

an accessible development of the mathematics underlying fractals and showing

how it may be applied in particular cases. The book covers a wide variety of

mathematical ideas that may be related to fractals, and, particularly in Part II,

ix

x Preface

provides a ﬂavour of what is available rather than exploring any one subject

in too much detail. The selection of topics is to some extent at the author’s

whim—there are certainly some important applications that are not included.

Some of the material dates back to early in the twentieth century whilst some is

very recent.

Notes and references are provided at the end of each chapter. The references

are by no means exhaustive, indeed complete references on the variety of topics

covered would ﬁll a large volume. However, it is hoped that enough information

is included to enable those who wish to do so to pursue any topic further.

It would be possible to use the book as a basis for a course on the mathe-

matics of fractals, at postgraduate or, perhaps, ﬁnal-year undergraduate level, and

exercises are included at the end of each chapter to facilitate this. Harder sections

and proofs are marked with an asterisk, and may be omitted without interrupting

the development.

An effort has been made to keep the mathematics to a level that can be under-

stood by a mathematics or physics graduate, and, for the most part, by a diligent

ﬁnal-year undergraduate. In particular, measure theoretic ideas have been kept to

a minimum, and the reader is encouraged to think of measures as ‘mass distribu-

tions’ on sets. Provided that it is accepted that measures with certain (intuitively

almost obvious) properties exist, there is little need for technical measure theory

in our development.

Results are always stated precisely to avoid the confusion which would other-

wise result. Our approach is generally rigorous, but some of the harder or more

technical proofs are either just sketched or omitted altogether. (However, a few

harder proofs that are not available in that form elsewhere have been included, in

particular those on sets with large intersection and on random fractals.) Suitable

diagrams can be a help in understanding the proofs, many of which are of a

geometric nature. Some diagrams are included in the book; the reader may ﬁnd

it helpful to draw others.

Chapter 1 begins with a rapid survey of some basic mathematical concepts

and notation, for example, from the theory of sets and functions, that are used

throughout the book. It also includes an introductory section on measure theory

and mass distributions which, it is hoped, will be found adequate. The section

on probability theory may be helpful for the chapters on random fractals and

Brownian motion.

With the wide variety of topics covered it is impossible to be entirely consistent

in use of notation and inevitably there sometimes has to be a compromise between

consistency within the book and standard usage.

In the last few years fractals have become enormously popular as an art form,

with the advent of computer graphics, and as a model of a wide variety of physical

phenomena. Whilst it is possible in some ways to appreciate fractals with little or

no knowledge of their mathematics, an understanding of the mathematics that can

be applied to such a diversity of objects certainly enhances one’s appreciation.

The phrase ‘the beauty of fractals’ is often heard—it is the author’s belief that

much of their beauty is to be found in their mathematics.

Preface xi

It is a pleasure to acknowledge those who have assisted in the preparation

of this book. Philip Drazin and Geoffrey Grimmett provided helpful comments

on parts of the manuscript. Peter Shiarly gave valuable help with the computer

drawings and Aidan Foss produced some diagrams. I am indebted to Charlotte

Farmer, Jackie Cowling and Stuart Gale of John Wiley and Sons for overseeing

the production of the book.

Special thanks are due to David Marsh—not only did he make many useful

comments on the manuscript and produce many of the computer pictures, but he

also typed the entire manuscript in a most expert way.

Finally, I would like to thank my wife Isobel for her support and encourage-

ment, which extended to reading various drafts of the book.

Kenneth J. Falconer

Bristol, April 1989

Preface to the second edition

It is thirteen years since Fractal Geometry—Mathematical Foundations and Appli-

cations was ﬁrst published. In the meantime, the mathematics and applications of

fractals have advanced enormously, with an ever-widening interest in the subject

at all levels. The book was originally written for those working in mathematics

and science who wished to know more about fractal mathematics. Over the past

few years, with changing interests and approaches to mathematics teaching, many

universities have introduced undergraduate and postgraduate courses on fractal

geometry, and a considerable number have been based on parts of this book.

Thus, this new edition has two main aims. First, it indicates some recent devel-

opments in the subject, with updated notes and suggestions for further reading.

Secondly, more attention is given to the needs of students using the book as a

course text, with extra details to help understanding, along with the inclusion of

further exercises.

Parts of the book have been rewritten. In particular, multifractal theory has

advanced considerably since the ﬁrst edition was published, so the chapter on

‘Multifractal Measures’ has been completely rewritten. The notes and references

have been updated. Numerous minor changes, corrections and additions have

been incorporated, and some of the notation and terminology has been changed to

conform with what has become standard usage. Many of the diagrams have been

replaced to take advantage of the more sophisticated computer technology now

available. Where possible, the numbering of sections, equations and ﬁgures has

been left as in the ﬁrst edition, so that earlier references to the book remain valid.

Further exercises have been added at the end of the chapters. Solutions to these

exercises and additional supplementary material may be found on the world wide

web at

http://www.wileyeurope.com/fractal

In 1997 a sequel, Techniques in Fractal Geometry, was published, presenting

a variety of techniques and ideas current in fractal research. Readers wishing

to study fractal mathematics beyond the bounds of this book may ﬁnd the

sequel helpful.

I am most grateful to all who have made constructive suggestions on the text. In

particular I am indebted to Carmen Fern

´

andez, Gwyneth Stallard and Alex Cain

xiii

xiv Preface to the second edition

for help with this revision. I am also very grateful for the continuing support

given to the book by the staff of John Wiley & Sons, and in particular to Rob

Calver and Lucy Bryan, for overseeing the production of this second edition and

John O’Connor and Louise Page for the cover design.

Kenneth J. Falconer

St Andrews, January 2003

Course suggestions

There is far too much material in this book for a standard length course on

fractal geometry. Depending on the emphasis required, appropriate sections may

be selected as a basis for an undergraduate or postgraduate course.

A course for mathematics students could be based on the following sections.

(a) Mathematical background

1.1 Basic set theory; 1.2 Functions and limits; 1.3 Measures and mass

distributions.

(b) Box-counting dimensions

3.1 Box-counting dimensions; 3.2 Properties of box-counting dimensions.

(c) Hausdorff measures and dimension

2.1 Hausdorff measure; 2.2 Hausdorff dimension; 2.3 Calculation of Haus-

dorff dimension; 4.1 Basic methods of calculating dimensions.

(d) Iterated function systems

9.1 Iterated function systems; 9.2 Dimensions of self-similar sets; 9.3 Some

variations; 10.2 Continued fraction examples.

(e) Graphs of functions

11.1 Dimensions of graphs, the Weierstrass function and self-afﬁne graphs.

(f) Dynamical systems

13.1 Repellers and iterated function systems; 13.2 The logistic map.

(g) Iteration of complex functions

14.1 Sketch of general theory of Julia sets; 14.2 The Mandelbrot set; 14.3

Julia sets of quadratic functions.

xv

Introduction

In the past, mathematics has been concerned largely with sets and functions to

which the methods of classical calculus can be applied. Sets or functions that

are not sufﬁciently smooth or regular have tended to be ignored as ‘pathological’

and not worthy of study. Certainly, they were regarded as individual curiosities

and only rarely were thought of as a class to which a general theory might be

applicable.

In recent years this attitude has changed. It has been realized that a great deal

can be said, and is worth saying, about the mathematics of non-smooth objects.

Moreover, irregular sets provide a much better representation of many natural

phenomena than do the ﬁgures of classical geometry. Fractal geometry provides

a general framework for the study of such irregular sets.

We begin by looking brieﬂy at a number of simple examples of fractals, and

note some of their features.

The middle third Cantor set is one of the best known and most easily con-

structed fractals; nevertheless it displays many typical fractal characteristics. It

is constructed from a unit interval by a sequence of deletion operations; see

ﬁgure 0.1. Let E

0

be the interval [0, 1]. (Recall that [a, b] denotes the set of real

numbers x such that a

x b.) Let E

1

be the set obtained by deleting the mid-

dle third of E

0

,sothatE

1

consists of the two intervals [0,

1

3

]and[

2

3

, 1]. Deleting

the middle thirds of these intervals gives E

2

; thus E

2

comprises the four intervals

[0,

1

9

], [

2

9

,

1

3

], [

2

3

,

7

9

], [

8

9

, 1]. We continue in this way, with E

k

obtained by delet-

ing the middle third of each interval in E

k−1

. Thus E

k

consists of 2

k

intervals

each of length 3

−k

.Themiddle third Cantor set F consists of the numbers that

are in E

k

for all k; mathematically, F is the intersection

∞

k=0

E

k

. The Cantor

set F may be thought of as the limit of the sequence of sets E

k

as k tends to

inﬁnity. It is obviously impossible to draw the set F itself, with its inﬁnitesimal

detail, so ‘pictures of F ’ tend to be pictures of one of the E

k

, which are a good

approximation to F when k is reasonably large; see ﬁgure 0.1.

At ﬁrst glance it might appear that we have removed so much of the interval

[0, 1] during the construction of F , that nothing remains. In fact, F is an inﬁnite

(and indeed uncountable) set, which contains inﬁnitely many numbers in every

neighbourhood of each of its points. The middle third Cantor set F consists

xvii

xviii Introduction

01

E

0

E

1

E

2

E

3

E

4

E

5

F

F

L

F

R

1

3

2

3

Figure 0.1 Construction of the middle third Cantor set F , by repeated removal of the

middle third of intervals. Note that F

L

and F

R

, the left and right parts of F , are copies

of F scaled by a factor

1

3

precisely of those numbers in [0, 1] whose base-3 expansion does not contain

the digit 1, i.e. all numbers a

1

3

−1

+ a

2

3

−2

+ a

3

3

−3

+···with a

i

= 0or2for

each i. To see this, note that to get E

1

from E

0

we remove those numbers with

a

1

= 1, to get E

2

from E

1

we remove those numbers with a

2

= 1, and so on.

We list some of the features of the middle third Cantor set F ;asweshallsee,

similar features are found in many fractals.

(i) F is self-similar. It is clear that the part of F in the interval [0,

1

3

]andthe

part of F in [

2

3

, 1] are both geometrically similar to F , scaled by a factor

1

3

. Again, the parts of F in each of the four intervals of E

2

are similar to

F but scaled by a factor

1

9

, and so on. The Cantor set contains copies of

itself at many different scales.

(ii) The set F has a ‘ﬁne structure’; that is, it contains detail at arbitrarily

small scales. The more we enlarge the picture of the Cantor set, the more

gaps become apparent to the eye.

(iii) Although F has an intricate detailed structure, the actual deﬁnition of F

is very straightforward.

(iv) F is obtained by a recursive procedure. Our construction consisted of

repeatedly removing the middle thirds of intervals. Successive steps give

increasingly good approximations E

k

to the set F .

(v) The geometry of F is not easily described in classical terms: it is not the

locus of the points that satisfy some simple geometric condition, nor is it

the set of solutions of any simple equation.

(vi) It is awkward to describe the local geometry of F —near each of its points

are a large number of other points, separated by gaps of varying lengths.

(vii) Although F is in some ways quite a large set (it is uncountably inﬁnite),

its size is not quantiﬁed by the usual measures such as length—by any

reasonable deﬁnition F has length zero.

Our second example, the von Koch curve, will also be familiar to many readers;

seeﬁgure0.2.WeletE

0

be a line segment of unit length. The set E

1

consists of

the four segments obtained by removing the middle third of E

0

and replacing it

Introduction xix

E

0

E

1

E

2

F

E

3

(a)

(b)

Figure 0.2 (a) Construction of the von Koch curve F . At each stage, the middle third of

each interval is replaced by the other two sides of an equilateral triangle. (b) Three von

Koch curves ﬁtted together to form a snowﬂake curve

by the other two sides of the equilateral triangle based on the removed segment.

We construct E

2

by applying the same procedure to each of the segments in E

1

,

and so on. Thus E

k

comes from replacing the middle third of each straight line

segment of E

k−1

by the other two sides of an equilateral triangle. When k is

xx Introduction

large, the curves E

k−1

and E

k

differ only in ﬁne detail and as k tends to inﬁnity,

the sequence of polygonal curves E

k

approaches a limiting curve F , called the

von Koch curve.

The von Koch curve has features in many ways similar to those listed for

the middle third Cantor set. It is made up of four ‘quarters’ each similar to the

whole, but scaled by a factor

1

3

. The ﬁne structure is reﬂected in the irregularities

at all scales; nevertheless, this intricate structure stems from a basically simple

construction. Whilst it is reasonable to call F a curve, it is much too irregular

to have tangents in the classical sense. A simple calculation shows that E

k

is of

length

4

3

k

; letting k tend to inﬁnity implies that F has inﬁnite length. On the

other hand, F occupies zero area in the plane, so neither length nor area provides

a very useful description of the size of F.

Many other sets may be constructed using such recursive procedures. For

example, the Sierpi´nski triangle or gasket is obtained by repeatedly removing

(inverted) equilateral triangles from an initial equilateral triangle of unit side-

length; see ﬁgure 0.3. (For many purposes, it is better to think of this procedure

as repeatedly replacing an equilateral triangle by three triangles of half the height.)

A plane analogue of the Cantor set, a ‘Cantor dust’, is illustrated in ﬁgure 0.4. At

each stage each remaining square is divided into 16 smaller squares of which four

are kept and the rest discarded. (Of course, other arrangements or numbers of

squares could be used to get different sets.) It should be clear that such examples

have properties similar to those mentioned in connection with the Cantor set and

the von Koch curve. The example depicted in ﬁgure 0.5 is constructed using two

different similarity ratios.

There are many other types of construction, some of which will be discussed

in detail later in the book, that also lead to sets with these sorts of properties.

E

0

E

1

F

E

2

Figure 0.3 Construction of the Sierpi

´

nski triangle (dim

H

F = dim

B

F = log 3/ log 2)

Introduction xxi

E

0

E

1

F

E

2

Figure 0.4 Construction of a ‘Cantor dust’ (dim

H

F = dim

B

F = 1)

E

0

E

1

F

E

2

Figure 0.5 Construction of a self-similar fractal with two different similarity ratios

xxii Introduction

The highly intricate structure of the Julia set illustrated in ﬁgure 0.6 stems from

the single quadratic function f(z)= z

2

+ c for a suitable constant c. Although

the set is not strictly self-similar in the sense that the Cantor set and von Koch

curve are, it is ‘quasi-self-similar’ in that arbitrarily small portions of the set can

be magniﬁed and then distorted smoothly to coincide with a large part of the set.

Figure 0.7 shows the graph of the function f(t)=

∞

k=0

(

3

2

)

−k/2

sin((

3

2

)

k

t);the

inﬁnite summation leads to the graph having a ﬁne structure, rather than being a

smooth curve to which classical calculus is applicable.

Some of these constructions may be ‘randomized’. Figure 0.8 shows a ‘random

von Koch curve’—a coin was tossed at each step in the construction to determine

on which side of the curve to place the new pair of line segments. This random

curve certainly has a ﬁne structure, but the strict self-similarity of the von Koch

curve has been replaced by a ‘statistical self-similarity’.

These are all examples of sets that are commonly referred to as fractals. (The

word ‘fractal’ was coined by Mandelbrot in his fundamental essay from the Latin

fractus, meaning broken, to describe objects that were too irregular to ﬁt into a

traditional geometrical setting.) Properties such as those listed for the Cantor set

are characteristic of fractals, and it is sets with such properties that we will have

in mind throughout the book. Certainly, any fractal worthy of the name will

have a ﬁne structure, i.e. detail at all scales. Many fractals have some degree of

self-similarity—they are made up of parts that resemble the whole in some way.

Sometimes, the resemblance may be weaker than strict geometrical similarity;

for example, the similarity may be approximate or statistical.

Methods of classical geometry and calculus are unsuited to studying frac-

tals and we need alternative techniques. The main tool of fractal geometry

is dimension in its many forms. We are familiar enough with the idea that a

Figure 0.6 A Julia set

Introduction xxiii

3f (t)

2

1

0

−1

−2

−3

0123456

t

Figure 0.7 Graph of f(t) =

∞

k=0

(

3

2

)

−k/2

sin((

3

2

)

k

t)

(smooth) curve is a 1-dimensional object and a surface is 2-dimensional. It is

less clear that, for many purposes, the Cantor set should be regarded as having

dimension log 2/ log 3 = 0.631 and the von Koch curve as having dimen-

sion log 4/ log 3 = 1.262 This latter number is, at least, consistent with the

von Koch curve being ‘larger than 1-dimensional’ (having inﬁnite length) and

‘smaller than 2-dimensional’ (having zero area).

Figure 0.8 A random version of the von Koch curve

xxiv Introduction

(a)

(b)

(c)

(d)

Figure 0.9 Division of certain sets into four parts. The parts are similar to the whole with

ratios:

1

4

for line segment (a);

1

2

for square (b);

1

9

for middle third Cantor set (c);

1

3

for

von Koch curve (d)

The following argument gives one (rather crude) interpretation of the meaning

of these ‘dimensions’ indicating how they reﬂect scaling properties and self-

similarity. As ﬁgure 0.9 indicates, a line segment is made up of four copies of

itself, scaled by a factor

1

4

. The segment has dimension −log 4/ log

1

4

= 1. A

square, however, is made up of four copies of itself scaled by a factor

1

2

(i.e.

with half the side length) and has dimension −log 4/ log

1

2

= 2.Inthesameway,

the von Koch curve is made up of four copies of itself scaled by a factor

1

3

,and

has dimension −log 4/ log

1

3

= log 4/ log 3, and the Cantor set may be regarded

as comprising four copies of itself scaled by a factor

1

9

and having dimension

−log 4/ log

1

9

= log 2/ log 3. In general, a set made up of m copies of itself scaled

by a factor r might be thought of as having dimension −log m/ log r. The number

obtained in this way is usually referred to as the similarity dimension of the set.

Unfortunately, similarity dimension is meaningful only for a relatively small

class of strictly self-similar sets. Nevertheless, there are other deﬁnitions of

dimension that are much more widely applicable. For example, Hausdorff dimen-

sion and the box-counting dimensions may be deﬁned for any sets, and, in

these four examples, may be shown to equal the similarity dimension. The early

chapters of the book are concerned with the deﬁnition and properties of Hausdorff

and other dimensions, along with methods for their calculation. Very roughly, a

dimension provides a description of how much space a set ﬁlls. It is a measure of

the prominence of the irregularities of a set when viewed at very small scales. A

dimension contains much information about the geometrical properties of a set.

A word of warning is appropriate at this point. It is possible to deﬁne the

‘dimension’ of a set in many ways, some satisfactory and others less so. It

is important to realize that different deﬁnitions may give different values of

Introduction xxv

dimension for the same set, and may also have very different properties. Incon-

sistent usage has sometimes led to considerable confusion. In particular, warning

lights ﬂash in my mind (as in the minds of other mathematicians) whenever the

term ‘fractal dimension’ is seen. Though some authors attach a precise meaning

to this, I have known others interpret it inconsistently in a single piece of work.

The reader should always be aware of the deﬁnition in use in any discussion.

In his original essay, Mandelbrot deﬁned a fractal to be a set with Haus-

dorff dimension strictly greater than its topological dimension. (The topological

dimension of a set is always an integer and is 0 if it is totally disconnected, 1 if

each point has arbitrarily small neighbourhoods with boundary of dimension 0,

and so on.) This deﬁnition proved to be unsatisfactory in that it excluded a num-

ber of sets that clearly ought to be regarded as fractals. Various other deﬁnitions

have been proposed, but they all seem to have this same drawback.

My personal feeling is that the deﬁnition of a ‘fractal’ should be regarded in

the same way as a biologist regards the deﬁnition of ‘life’. There is no hard and

fast deﬁnition, but just a list of properties characteristic of a living thing, such

as the ability to reproduce or to move or to exist to some extent independently

of the environment. Most living things have most of the characteristics on the

list, though there are living objects that are exceptions to each of them. In the

same way, it seems best to regard a fractal as a set that has properties such

as those listed below, rather than to look for a precise deﬁnition which will

almost certainly exclude some interesting cases. From the mathematician’s point

of view, this approach is no bad thing. It is difﬁcult to avoid developing properties

of dimension other than in a way that applies to ‘fractal’ and ‘non-fractal’ sets

alike. For ‘non-fractals’, however, such properties are of little interest—they are

generally almost obvious and could be obtained more easily by other methods.

When we refer to a set F as a fractal, therefore, we will typically have the

following in mind.

(i) F has a ﬁne structure, i.e. detail on arbitrarily small scales.

(ii) F is too irregular to be described in traditional geometrical language, both

locally and globally.

(iii) Often F has some form of self-similarity, perhaps approximate or statis-

tical.

(iv) Usually, the ‘fractal dimension’ of F (deﬁned in some way) is greater

than its topological dimension.

(v) In most cases of interest F is deﬁned in a very simple way, perhaps

recursively.

What can we say about the geometry of as diverse a class of objects as frac-

tals? Classical geometry gives us a clue. In Part I of this book we study certain

analogues of familiar geometrical properties in the fractal situation. The orthog-

onal projection, or ‘shadow’ of a circle in space onto a plane is, in general, an

ellipse. The fractal projection theorems tell us about the ‘shadows’ of a fractal.

For many purposes, a tangent provides a good local approximation to a circle.

GEOMETRY

Mathematical Foundations

and Applications

FractalGeometry:MathematicalFoundationsandApplication. Second Edition Kenneth Falconer

2003 John Wiley & Sons, Ltd ISBNs: 0-470-84861-8 (HB); 0-470-84862-6 (PB)

FRACTAL

GEOMETRY

Mathematical Foundations

and Applications

Second Edition

Kenneth Falconer

University of St Andrews, UK

Copyright 2003 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,

West Sussex PO19 8SQ, England

Telephone (+44) 1243 779777

Email (for orders and customer service enquiries): cs-books@wiley.co.uk

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Contents

Preface ix

Preface to the second edition xiii

Course suggestions xv

Introduction xvii

Notes and references xxvii

PART I FOUNDATIONS 1

Chapter 1 Mathematical background 3

1.1 Basic set theory 3

1.2 Functions and limits 6

1.3 Measures and mass distributions 11

1.4 Notes on probability theory 17

1.5 Notes and references 24

Exercises 25

Chapter 2 Hausdorff measure and dimension 27

2.1 Hausdorff measure 27

2.2 Hausdorff dimension 31

2.3 Calculation of Hausdorff dimension—simple examples 34

*2.4 Equivalent deﬁnitions of Hausdorff dimension 35

*2.5 Finer deﬁnitions of dimension 36

2.6 Notes and references 37

Exercises 37

Chapter 3 Alternative deﬁnitions of dimension 39

3.1 Box-counting dimensions 41

3.2 Properties and problems of box-counting dimension 47

v

vi Contents

*3.3 Modiﬁed box-counting dimensions 49

*3.4 Packing measures and dimensions 50

3.5 Some other deﬁnitions of dimension 53

3.6 Notes and references 57

Exercises 57

Chapter 4 Techniques for calculating dimensions 59

4.1 Basic methods 59

4.2 Subsets of ﬁnite measure 68

4.3 Potential theoretic methods 70

*4.4 Fourier transform methods 73

4.5 Notes and references 74

Exercises 74

Chapter 5 Local structure of fractals 76

5.1 Densities 76

5.2 Structure of 1-sets 80

5.3 Tangents to

s

-sets 84

5.4 Notes and references 89

Exercises 89

Chapter 6 Projections of fractals 90

6.1 Projections of arbitrary sets 90

6.2 Projections of

s

-sets of integral dimension 93

6.3 Projections of arbitrary sets of integral dimension 95

6.4 Notes and references 97

Exercises 97

Chapter 7 Products of fractals 99

7.1 Product formulae 99

7.2 Notes and references 107

Exercises 107

Chapter 8 Intersections of fractals 109

8.1 Intersection formulae for fractals 110

*8.2 Sets with large intersection 113

8.3 Notes and references 118

Exercises 119

PART II APPLICATIONS AND EXAMPLES 121

Chapter 9 Iterated function systems—self-similar and self-afﬁne sets 123

9.1 Iterated function systems 123

9.2 Dimensions of self-similar sets 128

vii

9.3 Some variations 135

9.4 Self-afﬁne sets 139

9.5 Applications to encoding images 145

9.6 Notes and references 148

Exercises 149

Chapter 10 Examples from number theory 151

10.1 Distribution of digits of numbers 151

10.2 Continued fractions 153

10.3 Diophantine approximation 154

10.4 Notes and references 158

Exercises 158

Chapter 11 Graphs of functions 160

11.1 Dimensions of graphs 160

*11.2 Autocorrelation of fractal functions 169

11.3 Notes and references 173

Exercises 173

Chapter 12 Examples from pure mathematics 176

12.1 Duality and the Kakeya problem 176

12.2 Vitushkin’s conjecture 179

12.3 Convex functions 181

12.4 Groups and rings of fractional dimension 182

12.5 Notes and references 184

Exercises 185

Chapter 13 Dynamical systems 186

13.1 Repellers and iterated function systems 187

13.2 The logistic map 189

13.3 Stretching and folding transformations 193

13.4 The solenoid 198

13.5 Continuous dynamical systems 201

*13.6 Small divisor theory 205

*13.7 Liapounov exponents and entropies 208

13.8 Notes and references 211

Exercises 212

Chapter 14 Iteration of complex functions—Julia sets 215

14.1 General theory of Julia sets 215

14.2 Quadratic functions— the Mandelbrot set 223

14.3 Julia sets of quadratic functions 227

14.4 Characterization of quasi-circles by dimension 235

14.5 Newton’s method for solving polynomial equations 237

14.6 Notes and references 241

Exercises 242

viii Contents

Chapter 15 Random fractals 244

15.1 A random Cantor set 246

15.2 Fractal percolation 251

15.3 Notes and references 255

Exercises 256

Chapter 16 Brownian motion and Brownian surfaces 258

16.1 Brownian motion 258

16.2 Fractional Brownian motion 267

16.3 L

´

evy stable processes 271

16.4 Fractional Brownian surfaces 273

16.5 Notes and references 275

Exercises 276

Chapter 17 Multifractal measures 277

17.1 Coarse multifractal analysis 278

17.2 Fine multifractal analysis 283

17.3 Self-similar multifractals 286

17.4 Notes and references 296

Exercises 296

Chapter 18 Physical applications 298

18.1 Fractal growth 300

18.2 Singularities of electrostatic and gravitational potentials 306

18.3 Fluid dynamics and turbulence 307

18.4 Fractal antennas 309

18.5 Fractals in ﬁnance 311

18.6 Notes and references 315

Exercises 316

References 317

Index

329

Preface

I am frequently asked questions such as ‘What are fractals?’, ‘What is fractal

dimension?’, ‘How can one ﬁnd the dimension of a fractal and what does it

tell us anyway?’ or ‘How can mathematics be applied to fractals?’ This book

endeavours to answer some of these questions.

The main aim of the book is to provide a treatment of the mathematics asso-

ciated with fractals and dimensions at a level which is reasonably accessible to

those who encounter fractals in mathematics or science. Although basically a

mathematics book, it attempts to provide an intuitive as well as a mathematical

insight into the subject.

The book falls naturally into two parts. Part I is concerned with the general

theory of fractals and their geometry. Firstly, various notions of dimension and

methods for their calculation are introduced. Then geometrical properties of frac-

tals are investigated in much the same way as one might study the geometry of

classical ﬁgures such as circles or ellipses: locally a circle may be approximated

by a line segment, the projection or ‘shadow’ of a circle is generally an ellipse,

a circle typically intersects a straight line segment in two points (if at all), and

so on. There are fractal analogues of such properties, usually with dimension

playing a key r

ˆ

ole. Thus we consider, for example, the local form of fractals,

and projections and intersections of fractals.

Part II of the book contains examples of fractals, to which the theory of the

ﬁrst part may be applied, drawn from a wide variety of areas of mathematics

and physics. Topics include self-similar and self-afﬁne sets, graphs of functions,

examples from number theory and pure mathematics, dynamical systems, Julia

sets, random fractals and some physical applications.

There are many diagrams in the text and frequent illustrative examples. Com-

puter drawings of a variety of fractals are included, and it is hoped that enough

information is provided to enable readers with a knowledge of programming to

produce further drawings for themselves.

It is hoped that the book will be a useful reference for researchers, providing

an accessible development of the mathematics underlying fractals and showing

how it may be applied in particular cases. The book covers a wide variety of

mathematical ideas that may be related to fractals, and, particularly in Part II,

ix

x Preface

provides a ﬂavour of what is available rather than exploring any one subject

in too much detail. The selection of topics is to some extent at the author’s

whim—there are certainly some important applications that are not included.

Some of the material dates back to early in the twentieth century whilst some is

very recent.

Notes and references are provided at the end of each chapter. The references

are by no means exhaustive, indeed complete references on the variety of topics

covered would ﬁll a large volume. However, it is hoped that enough information

is included to enable those who wish to do so to pursue any topic further.

It would be possible to use the book as a basis for a course on the mathe-

matics of fractals, at postgraduate or, perhaps, ﬁnal-year undergraduate level, and

exercises are included at the end of each chapter to facilitate this. Harder sections

and proofs are marked with an asterisk, and may be omitted without interrupting

the development.

An effort has been made to keep the mathematics to a level that can be under-

stood by a mathematics or physics graduate, and, for the most part, by a diligent

ﬁnal-year undergraduate. In particular, measure theoretic ideas have been kept to

a minimum, and the reader is encouraged to think of measures as ‘mass distribu-

tions’ on sets. Provided that it is accepted that measures with certain (intuitively

almost obvious) properties exist, there is little need for technical measure theory

in our development.

Results are always stated precisely to avoid the confusion which would other-

wise result. Our approach is generally rigorous, but some of the harder or more

technical proofs are either just sketched or omitted altogether. (However, a few

harder proofs that are not available in that form elsewhere have been included, in

particular those on sets with large intersection and on random fractals.) Suitable

diagrams can be a help in understanding the proofs, many of which are of a

geometric nature. Some diagrams are included in the book; the reader may ﬁnd

it helpful to draw others.

Chapter 1 begins with a rapid survey of some basic mathematical concepts

and notation, for example, from the theory of sets and functions, that are used

throughout the book. It also includes an introductory section on measure theory

and mass distributions which, it is hoped, will be found adequate. The section

on probability theory may be helpful for the chapters on random fractals and

Brownian motion.

With the wide variety of topics covered it is impossible to be entirely consistent

in use of notation and inevitably there sometimes has to be a compromise between

consistency within the book and standard usage.

In the last few years fractals have become enormously popular as an art form,

with the advent of computer graphics, and as a model of a wide variety of physical

phenomena. Whilst it is possible in some ways to appreciate fractals with little or

no knowledge of their mathematics, an understanding of the mathematics that can

be applied to such a diversity of objects certainly enhances one’s appreciation.

The phrase ‘the beauty of fractals’ is often heard—it is the author’s belief that

much of their beauty is to be found in their mathematics.

Preface xi

It is a pleasure to acknowledge those who have assisted in the preparation

of this book. Philip Drazin and Geoffrey Grimmett provided helpful comments

on parts of the manuscript. Peter Shiarly gave valuable help with the computer

drawings and Aidan Foss produced some diagrams. I am indebted to Charlotte

Farmer, Jackie Cowling and Stuart Gale of John Wiley and Sons for overseeing

the production of the book.

Special thanks are due to David Marsh—not only did he make many useful

comments on the manuscript and produce many of the computer pictures, but he

also typed the entire manuscript in a most expert way.

Finally, I would like to thank my wife Isobel for her support and encourage-

ment, which extended to reading various drafts of the book.

Kenneth J. Falconer

Bristol, April 1989

Preface to the second edition

It is thirteen years since Fractal Geometry—Mathematical Foundations and Appli-

cations was ﬁrst published. In the meantime, the mathematics and applications of

fractals have advanced enormously, with an ever-widening interest in the subject

at all levels. The book was originally written for those working in mathematics

and science who wished to know more about fractal mathematics. Over the past

few years, with changing interests and approaches to mathematics teaching, many

universities have introduced undergraduate and postgraduate courses on fractal

geometry, and a considerable number have been based on parts of this book.

Thus, this new edition has two main aims. First, it indicates some recent devel-

opments in the subject, with updated notes and suggestions for further reading.

Secondly, more attention is given to the needs of students using the book as a

course text, with extra details to help understanding, along with the inclusion of

further exercises.

Parts of the book have been rewritten. In particular, multifractal theory has

advanced considerably since the ﬁrst edition was published, so the chapter on

‘Multifractal Measures’ has been completely rewritten. The notes and references

have been updated. Numerous minor changes, corrections and additions have

been incorporated, and some of the notation and terminology has been changed to

conform with what has become standard usage. Many of the diagrams have been

replaced to take advantage of the more sophisticated computer technology now

available. Where possible, the numbering of sections, equations and ﬁgures has

been left as in the ﬁrst edition, so that earlier references to the book remain valid.

Further exercises have been added at the end of the chapters. Solutions to these

exercises and additional supplementary material may be found on the world wide

web at

http://www.wileyeurope.com/fractal

In 1997 a sequel, Techniques in Fractal Geometry, was published, presenting

a variety of techniques and ideas current in fractal research. Readers wishing

to study fractal mathematics beyond the bounds of this book may ﬁnd the

sequel helpful.

I am most grateful to all who have made constructive suggestions on the text. In

particular I am indebted to Carmen Fern

´

andez, Gwyneth Stallard and Alex Cain

xiii

xiv Preface to the second edition

for help with this revision. I am also very grateful for the continuing support

given to the book by the staff of John Wiley & Sons, and in particular to Rob

Calver and Lucy Bryan, for overseeing the production of this second edition and

John O’Connor and Louise Page for the cover design.

Kenneth J. Falconer

St Andrews, January 2003

Course suggestions

There is far too much material in this book for a standard length course on

fractal geometry. Depending on the emphasis required, appropriate sections may

be selected as a basis for an undergraduate or postgraduate course.

A course for mathematics students could be based on the following sections.

(a) Mathematical background

1.1 Basic set theory; 1.2 Functions and limits; 1.3 Measures and mass

distributions.

(b) Box-counting dimensions

3.1 Box-counting dimensions; 3.2 Properties of box-counting dimensions.

(c) Hausdorff measures and dimension

2.1 Hausdorff measure; 2.2 Hausdorff dimension; 2.3 Calculation of Haus-

dorff dimension; 4.1 Basic methods of calculating dimensions.

(d) Iterated function systems

9.1 Iterated function systems; 9.2 Dimensions of self-similar sets; 9.3 Some

variations; 10.2 Continued fraction examples.

(e) Graphs of functions

11.1 Dimensions of graphs, the Weierstrass function and self-afﬁne graphs.

(f) Dynamical systems

13.1 Repellers and iterated function systems; 13.2 The logistic map.

(g) Iteration of complex functions

14.1 Sketch of general theory of Julia sets; 14.2 The Mandelbrot set; 14.3

Julia sets of quadratic functions.

xv

Introduction

In the past, mathematics has been concerned largely with sets and functions to

which the methods of classical calculus can be applied. Sets or functions that

are not sufﬁciently smooth or regular have tended to be ignored as ‘pathological’

and not worthy of study. Certainly, they were regarded as individual curiosities

and only rarely were thought of as a class to which a general theory might be

applicable.

In recent years this attitude has changed. It has been realized that a great deal

can be said, and is worth saying, about the mathematics of non-smooth objects.

Moreover, irregular sets provide a much better representation of many natural

phenomena than do the ﬁgures of classical geometry. Fractal geometry provides

a general framework for the study of such irregular sets.

We begin by looking brieﬂy at a number of simple examples of fractals, and

note some of their features.

The middle third Cantor set is one of the best known and most easily con-

structed fractals; nevertheless it displays many typical fractal characteristics. It

is constructed from a unit interval by a sequence of deletion operations; see

ﬁgure 0.1. Let E

0

be the interval [0, 1]. (Recall that [a, b] denotes the set of real

numbers x such that a

x b.) Let E

1

be the set obtained by deleting the mid-

dle third of E

0

,sothatE

1

consists of the two intervals [0,

1

3

]and[

2

3

, 1]. Deleting

the middle thirds of these intervals gives E

2

; thus E

2

comprises the four intervals

[0,

1

9

], [

2

9

,

1

3

], [

2

3

,

7

9

], [

8

9

, 1]. We continue in this way, with E

k

obtained by delet-

ing the middle third of each interval in E

k−1

. Thus E

k

consists of 2

k

intervals

each of length 3

−k

.Themiddle third Cantor set F consists of the numbers that

are in E

k

for all k; mathematically, F is the intersection

∞

k=0

E

k

. The Cantor

set F may be thought of as the limit of the sequence of sets E

k

as k tends to

inﬁnity. It is obviously impossible to draw the set F itself, with its inﬁnitesimal

detail, so ‘pictures of F ’ tend to be pictures of one of the E

k

, which are a good

approximation to F when k is reasonably large; see ﬁgure 0.1.

At ﬁrst glance it might appear that we have removed so much of the interval

[0, 1] during the construction of F , that nothing remains. In fact, F is an inﬁnite

(and indeed uncountable) set, which contains inﬁnitely many numbers in every

neighbourhood of each of its points. The middle third Cantor set F consists

xvii

xviii Introduction

01

E

0

E

1

E

2

E

3

E

4

E

5

F

F

L

F

R

1

3

2

3

Figure 0.1 Construction of the middle third Cantor set F , by repeated removal of the

middle third of intervals. Note that F

L

and F

R

, the left and right parts of F , are copies

of F scaled by a factor

1

3

precisely of those numbers in [0, 1] whose base-3 expansion does not contain

the digit 1, i.e. all numbers a

1

3

−1

+ a

2

3

−2

+ a

3

3

−3

+···with a

i

= 0or2for

each i. To see this, note that to get E

1

from E

0

we remove those numbers with

a

1

= 1, to get E

2

from E

1

we remove those numbers with a

2

= 1, and so on.

We list some of the features of the middle third Cantor set F ;asweshallsee,

similar features are found in many fractals.

(i) F is self-similar. It is clear that the part of F in the interval [0,

1

3

]andthe

part of F in [

2

3

, 1] are both geometrically similar to F , scaled by a factor

1

3

. Again, the parts of F in each of the four intervals of E

2

are similar to

F but scaled by a factor

1

9

, and so on. The Cantor set contains copies of

itself at many different scales.

(ii) The set F has a ‘ﬁne structure’; that is, it contains detail at arbitrarily

small scales. The more we enlarge the picture of the Cantor set, the more

gaps become apparent to the eye.

(iii) Although F has an intricate detailed structure, the actual deﬁnition of F

is very straightforward.

(iv) F is obtained by a recursive procedure. Our construction consisted of

repeatedly removing the middle thirds of intervals. Successive steps give

increasingly good approximations E

k

to the set F .

(v) The geometry of F is not easily described in classical terms: it is not the

locus of the points that satisfy some simple geometric condition, nor is it

the set of solutions of any simple equation.

(vi) It is awkward to describe the local geometry of F —near each of its points

are a large number of other points, separated by gaps of varying lengths.

(vii) Although F is in some ways quite a large set (it is uncountably inﬁnite),

its size is not quantiﬁed by the usual measures such as length—by any

reasonable deﬁnition F has length zero.

Our second example, the von Koch curve, will also be familiar to many readers;

seeﬁgure0.2.WeletE

0

be a line segment of unit length. The set E

1

consists of

the four segments obtained by removing the middle third of E

0

and replacing it

Introduction xix

E

0

E

1

E

2

F

E

3

(a)

(b)

Figure 0.2 (a) Construction of the von Koch curve F . At each stage, the middle third of

each interval is replaced by the other two sides of an equilateral triangle. (b) Three von

Koch curves ﬁtted together to form a snowﬂake curve

by the other two sides of the equilateral triangle based on the removed segment.

We construct E

2

by applying the same procedure to each of the segments in E

1

,

and so on. Thus E

k

comes from replacing the middle third of each straight line

segment of E

k−1

by the other two sides of an equilateral triangle. When k is

xx Introduction

large, the curves E

k−1

and E

k

differ only in ﬁne detail and as k tends to inﬁnity,

the sequence of polygonal curves E

k

approaches a limiting curve F , called the

von Koch curve.

The von Koch curve has features in many ways similar to those listed for

the middle third Cantor set. It is made up of four ‘quarters’ each similar to the

whole, but scaled by a factor

1

3

. The ﬁne structure is reﬂected in the irregularities

at all scales; nevertheless, this intricate structure stems from a basically simple

construction. Whilst it is reasonable to call F a curve, it is much too irregular

to have tangents in the classical sense. A simple calculation shows that E

k

is of

length

4

3

k

; letting k tend to inﬁnity implies that F has inﬁnite length. On the

other hand, F occupies zero area in the plane, so neither length nor area provides

a very useful description of the size of F.

Many other sets may be constructed using such recursive procedures. For

example, the Sierpi´nski triangle or gasket is obtained by repeatedly removing

(inverted) equilateral triangles from an initial equilateral triangle of unit side-

length; see ﬁgure 0.3. (For many purposes, it is better to think of this procedure

as repeatedly replacing an equilateral triangle by three triangles of half the height.)

A plane analogue of the Cantor set, a ‘Cantor dust’, is illustrated in ﬁgure 0.4. At

each stage each remaining square is divided into 16 smaller squares of which four

are kept and the rest discarded. (Of course, other arrangements or numbers of

squares could be used to get different sets.) It should be clear that such examples

have properties similar to those mentioned in connection with the Cantor set and

the von Koch curve. The example depicted in ﬁgure 0.5 is constructed using two

different similarity ratios.

There are many other types of construction, some of which will be discussed

in detail later in the book, that also lead to sets with these sorts of properties.

E

0

E

1

F

E

2

Figure 0.3 Construction of the Sierpi

´

nski triangle (dim

H

F = dim

B

F = log 3/ log 2)

Introduction xxi

E

0

E

1

F

E

2

Figure 0.4 Construction of a ‘Cantor dust’ (dim

H

F = dim

B

F = 1)

E

0

E

1

F

E

2

Figure 0.5 Construction of a self-similar fractal with two different similarity ratios

xxii Introduction

The highly intricate structure of the Julia set illustrated in ﬁgure 0.6 stems from

the single quadratic function f(z)= z

2

+ c for a suitable constant c. Although

the set is not strictly self-similar in the sense that the Cantor set and von Koch

curve are, it is ‘quasi-self-similar’ in that arbitrarily small portions of the set can

be magniﬁed and then distorted smoothly to coincide with a large part of the set.

Figure 0.7 shows the graph of the function f(t)=

∞

k=0

(

3

2

)

−k/2

sin((

3

2

)

k

t);the

inﬁnite summation leads to the graph having a ﬁne structure, rather than being a

smooth curve to which classical calculus is applicable.

Some of these constructions may be ‘randomized’. Figure 0.8 shows a ‘random

von Koch curve’—a coin was tossed at each step in the construction to determine

on which side of the curve to place the new pair of line segments. This random

curve certainly has a ﬁne structure, but the strict self-similarity of the von Koch

curve has been replaced by a ‘statistical self-similarity’.

These are all examples of sets that are commonly referred to as fractals. (The

word ‘fractal’ was coined by Mandelbrot in his fundamental essay from the Latin

fractus, meaning broken, to describe objects that were too irregular to ﬁt into a

traditional geometrical setting.) Properties such as those listed for the Cantor set

are characteristic of fractals, and it is sets with such properties that we will have

in mind throughout the book. Certainly, any fractal worthy of the name will

have a ﬁne structure, i.e. detail at all scales. Many fractals have some degree of

self-similarity—they are made up of parts that resemble the whole in some way.

Sometimes, the resemblance may be weaker than strict geometrical similarity;

for example, the similarity may be approximate or statistical.

Methods of classical geometry and calculus are unsuited to studying frac-

tals and we need alternative techniques. The main tool of fractal geometry

is dimension in its many forms. We are familiar enough with the idea that a

Figure 0.6 A Julia set

Introduction xxiii

3f (t)

2

1

0

−1

−2

−3

0123456

t

Figure 0.7 Graph of f(t) =

∞

k=0

(

3

2

)

−k/2

sin((

3

2

)

k

t)

(smooth) curve is a 1-dimensional object and a surface is 2-dimensional. It is

less clear that, for many purposes, the Cantor set should be regarded as having

dimension log 2/ log 3 = 0.631 and the von Koch curve as having dimen-

sion log 4/ log 3 = 1.262 This latter number is, at least, consistent with the

von Koch curve being ‘larger than 1-dimensional’ (having inﬁnite length) and

‘smaller than 2-dimensional’ (having zero area).

Figure 0.8 A random version of the von Koch curve

xxiv Introduction

(a)

(b)

(c)

(d)

Figure 0.9 Division of certain sets into four parts. The parts are similar to the whole with

ratios:

1

4

for line segment (a);

1

2

for square (b);

1

9

for middle third Cantor set (c);

1

3

for

von Koch curve (d)

The following argument gives one (rather crude) interpretation of the meaning

of these ‘dimensions’ indicating how they reﬂect scaling properties and self-

similarity. As ﬁgure 0.9 indicates, a line segment is made up of four copies of

itself, scaled by a factor

1

4

. The segment has dimension −log 4/ log

1

4

= 1. A

square, however, is made up of four copies of itself scaled by a factor

1

2

(i.e.

with half the side length) and has dimension −log 4/ log

1

2

= 2.Inthesameway,

the von Koch curve is made up of four copies of itself scaled by a factor

1

3

,and

has dimension −log 4/ log

1

3

= log 4/ log 3, and the Cantor set may be regarded

as comprising four copies of itself scaled by a factor

1

9

and having dimension

−log 4/ log

1

9

= log 2/ log 3. In general, a set made up of m copies of itself scaled

by a factor r might be thought of as having dimension −log m/ log r. The number

obtained in this way is usually referred to as the similarity dimension of the set.

Unfortunately, similarity dimension is meaningful only for a relatively small

class of strictly self-similar sets. Nevertheless, there are other deﬁnitions of

dimension that are much more widely applicable. For example, Hausdorff dimen-

sion and the box-counting dimensions may be deﬁned for any sets, and, in

these four examples, may be shown to equal the similarity dimension. The early

chapters of the book are concerned with the deﬁnition and properties of Hausdorff

and other dimensions, along with methods for their calculation. Very roughly, a

dimension provides a description of how much space a set ﬁlls. It is a measure of

the prominence of the irregularities of a set when viewed at very small scales. A

dimension contains much information about the geometrical properties of a set.

A word of warning is appropriate at this point. It is possible to deﬁne the

‘dimension’ of a set in many ways, some satisfactory and others less so. It

is important to realize that different deﬁnitions may give different values of

Introduction xxv

dimension for the same set, and may also have very different properties. Incon-

sistent usage has sometimes led to considerable confusion. In particular, warning

lights ﬂash in my mind (as in the minds of other mathematicians) whenever the

term ‘fractal dimension’ is seen. Though some authors attach a precise meaning

to this, I have known others interpret it inconsistently in a single piece of work.

The reader should always be aware of the deﬁnition in use in any discussion.

In his original essay, Mandelbrot deﬁned a fractal to be a set with Haus-

dorff dimension strictly greater than its topological dimension. (The topological

dimension of a set is always an integer and is 0 if it is totally disconnected, 1 if

each point has arbitrarily small neighbourhoods with boundary of dimension 0,

and so on.) This deﬁnition proved to be unsatisfactory in that it excluded a num-

ber of sets that clearly ought to be regarded as fractals. Various other deﬁnitions

have been proposed, but they all seem to have this same drawback.

My personal feeling is that the deﬁnition of a ‘fractal’ should be regarded in

the same way as a biologist regards the deﬁnition of ‘life’. There is no hard and

fast deﬁnition, but just a list of properties characteristic of a living thing, such

as the ability to reproduce or to move or to exist to some extent independently

of the environment. Most living things have most of the characteristics on the

list, though there are living objects that are exceptions to each of them. In the

same way, it seems best to regard a fractal as a set that has properties such

as those listed below, rather than to look for a precise deﬁnition which will

almost certainly exclude some interesting cases. From the mathematician’s point

of view, this approach is no bad thing. It is difﬁcult to avoid developing properties

of dimension other than in a way that applies to ‘fractal’ and ‘non-fractal’ sets

alike. For ‘non-fractals’, however, such properties are of little interest—they are

generally almost obvious and could be obtained more easily by other methods.

When we refer to a set F as a fractal, therefore, we will typically have the

following in mind.

(i) F has a ﬁne structure, i.e. detail on arbitrarily small scales.

(ii) F is too irregular to be described in traditional geometrical language, both

locally and globally.

(iii) Often F has some form of self-similarity, perhaps approximate or statis-

tical.

(iv) Usually, the ‘fractal dimension’ of F (deﬁned in some way) is greater

than its topological dimension.

(v) In most cases of interest F is deﬁned in a very simple way, perhaps

recursively.

What can we say about the geometry of as diverse a class of objects as frac-

tals? Classical geometry gives us a clue. In Part I of this book we study certain

analogues of familiar geometrical properties in the fractal situation. The orthog-

onal projection, or ‘shadow’ of a circle in space onto a plane is, in general, an

ellipse. The fractal projection theorems tell us about the ‘shadows’ of a fractal.

For many purposes, a tangent provides a good local approximation to a circle.

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