An Invitation to Fractal Geometry and Its Applications

An Invitation to Fractal

Geometry and Its Applications

Michel L. Lapidus, Dana Clahane, Robert G. Niemeyer

University of California, Riverside

Department of Mathematics

900 Big Springs Rd.

Riverside, CA, 92521

[email protected], [email protected], [email protected]

(Draft Version)

October 7, 2006

Contents

1 Introduction 7

2 Preliminaries 9

2.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Set-Theoretic Notation and Concepts . . . . . . . . . . . . . . . . 16

2.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Point-Set Topology and Metric Spaces . . . . . . . . . . . . . . . 23

3 Concepts in Fractal Geometry 25

3.1 Euclidean Generators . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Recursion and Fractal Construction . . . . . . . . . . . . . . . . 31

3.3 Self-Similar and Self-Affinity . . . . . . . . . . . . . . . . . . . . . 36

3.4 Infinite Length and Uncountability . . . . . . . . . . . . . . . . . 45

3.5 Non-Integer Dimension . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Various Fractal Dimensions . . . . . . . . . . . . . . . . . . . . . 51

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3

4 CONTENTS

4 Measure Theory 53

5 Dimension theory (intuitive approach...box counting and Haus-

dorff) 57

5.0.1 Self-Similarity Dimension . . . . . . . . . . . . . . . . . . 58

5.0.2 Box-Counting Dimension . . . . . . . . . . . . . . . . . . 61

6 Iterated Function Systems 63

7 Fractal Curves and Sets 65

7.1 Fractal Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.2 Taming the Mathematical Monsters . . . . . . . . . . . . . . . . 68

7.2.1 Nondifferentiability of the Weierstrass Function . . . . . . 70

7.3 Definitions for a Fractal (many competing ones and the definitions) 78

8 Fractal Strings 79

9 Calculus on Fractals (i.e. lite analysis on fractals) 81

10 Applications 83

10.1 Brownian Motions . . . . . . . . . . . . . . . . . . . . . . . . . . 83

10.1.1 Theoretical Interpretation . . . . . . . . . . . . . . . . . . 84

10.1.2 Fractal Properties of Brownian Paths & Graphs . . . . . 92

10.2 Fractal Economics (Mandelbrot’s work) . . . . . . . . . . . . . . 95

10.3 Fractal Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

10.4 Fractals in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 95

CONTENTS 5

10.5 Fractals in cognition,sociology (Dr. Lapidus has a book on shelf) 95

10.6 Fractal Drums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

10.7 Fractal Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

• Historical significance

• Intuitive motivation

• Intuitive motivation for fractals

• Outline of chapter

Chapter 3

Concepts in Fractal

Geometry

Fractal Geometry has its roots in a variety of analytical problems. Many of the

standard sets found in every fractal geometer’s tool box were originally discussed

in the context of counter-examples to long-standing problems. In some cases,

the development of fractal sets were in answer to physical problems.

The early work of Fourier on heat diffusion and harmonic analysis motivated

Riemann, Weierstrass and others to investigate the analytic nature of explicit

trigonometric series. As a result of Riemann and Weierstrass’ separate works, it

was settled once and for all that highly nondifferentiable and entirely nondiffer-

entiable functions did exist. In fact, the mathematical community was quickly,

though begrudgingly, conceding that nondifferentiable functions were the norm

25

26 Chapter 3. Concepts in Fractal Geometry

and differentiable functions were a special case. But this was just the beginning

of a deep mathematical mystery.

Cantor was a contemporary of Weierstrass and did much to create an up-

heaval in mathematics, as well. In particular, his discoveries led to a recon-

ceptualization of similarity. While his original results were concerned with Set

Theory and Harmonic Analysis, his name is now pervasive in many mathemati-

cal branches. In fact, his famous Cantor set is the first “monster” many students

are introduced to, mainly because it is a perfect set with Lebesgue measure zero

and cardinality equal to that of the unit interval.

The generation after Weierstrass and Cantor included Giuseppe Peano, Helge

von Koch, Waclaw Sierpinski, and others, each having a surprise of their own.

Peano shook the geometric intuition of every mathematician when he demon-

strated that there exists a continuous map f from the unit interval I to the unit

square I × I whose image fills the space I × I.

Von Koch’s example of a nowhere differentiable function was borne out of a

desire to have a more natural example of Weierstrass’ findings.

Waclaw Sierpinski, while known to budding fractal geometers for his Sierpin-

ski gasket made an astonishing discovery with the Sierpinski carpet. He found

that any planar Jordan curve could be embedded homeomorphically into the

Sierpinski carpet. Though this fact did not hold for the Sierpinski gasket, Karl

Menger was able to extend the results to higher dimensions.

Around the time of Sierpinski and Menger, Wiener was placing the work of

Robert Brown, Albert Einstein and others on a rigorous foundation. Brownian

27

paths and motion, as formalized by Wiener were nondifferentiable paths and

physical Brownian motion was the expression of a naturally occurring statisti-

cally self-similar fractal.

For years, physicists have been developing methods to explain geometric

phenomena that were not fitting well with traditional Euclidean concepts. As

a field, Fractal Geometry cannot be described as a purely theoretical study

of fractal sets, nor can insight be gained by rephrasing it as ‘the geometry of

fractals,’ if such a concept were even rigorously defined. Nor is it an area entirely

devoted to explaining roughness in Nature. Rather, Fractal Geometry is the

unification of seemingly unrelated fields under an umbrella of pure mathematics

that ties together the various concepts in the physical sciences by providing a

rigorous mathematical foundation.

More recent motivation for developing the study of fractal sets came from

Benoit B. Mandelbrot’s paper, How long is the coast of Britain [16], and the

culmination of his work in the treatise, The Fractal Geometry of Nature [13]. In

such works, the nature of roughness was closely examined and placed on a more

rigorous foundation. For example, the shoreline seems rather smooth at large

scales, but any attempt to measure the coast shows that finer and finer scales

only illuminate the shore’s roughness and make the length longer and longer.

Topologically speaking, a fractal and non-fractal curve are equivalent. The

first attempt at developing a method for discerning was made by Felix Hausdorff.

Though Hausdorff is a name traditionally associated with particular topological

spaces, the topologically noninvariant method developed by Hausdorff entailed


assigning a value to a set that indicated a point at which the measure of a set

jumped from infinity to zero. As part of the goal of research, mathematicians

now attempt to classify sets in a non-topological manner so as to provide a well-

defined method for differentiating between two topologically equivalent sets.

For over forty years, mathematicians have been confronted with the problem

of what constitutes a fractal set. The word fractal, having its origins in the Latin

word for broken - fractus - was coined by Benoit Mandelbrot. The following

fractal criteria was set forth by Mandelbrot.

Criteria 3.0.1 ( [13]). “A fractal is, by definition, a set for which the Hausdorff–

Besicovitch dimension strictly exceeds the topological dimension.”

This definition was later dismissed by Mandelbrot as being insufficient in its

ability to describe a larger class of sets that the intellectual community accepted

as being fractal. In particular, this definition excluded the Devil’s Staircase.

Hence, Mandelbrot next proposed that a fractal be defined as follows:

Criteria 3.0.2 ( [17]). A fractal is a shape made of parts similar to the whole

in some way.

Researchers across the various scientific disciplines readily agreed that this

was a better definition for fractal, but was still inadequate for rigorous mathe-

matical discussion (see [14]).

One main problem in the discipline thus far is the lack of an official rigorous

definition for fractal. While research in this area will be discussed and analyzed,

we shall first introduce the various criteria used to decide whether or not a set

3.1. Euclidean Generators 29

is fractal.

Before we discuss the various fractal criteria, it is extremely important to

realize that a fractal set F cannot be illustrated in reality. This stems from

the fact that Nature is discrete. Eventually the granular nature of matter rears

its ugly head and shows that everything is not smooth at some fixed scale.

For example, a smooth curve drawn on paper will exhibit a high degree of

roughness at the microscopic scale. Another example is the fern. At some level,

the fern ceases to exist as a ‘self-repeating’ structure and the cellular structure

becomes apparent. Hence, the fern has a limited self-similar structure. What

is important is that the abstract knowledge gained in the study of what are

termed self-similar sets offers insight into finitely self-similar structures.

3.1 Euclidean Generators

Traditional Euclidean shapes are points (a1, a2, ..., an), lines #»r , n-spheres, and

n-dimensional polyhedra. Such shapes are finite in their measure and can be

unioned and intersected in such a way so as to produce more complicated Eu-

clidean shapes. Furthermore, Euclidean shapes, are used in the construction of

fractal sets. However, this does not imply that a fractal is merely a complicated

Euclidean shape in disguise.

A fractal can often be constructed by a particular iterative process that

begins with an initiator and/or generator. The initiator and generator are rep-

resented by the familiar Euclidean shapes, i.e. line segments, circles, polygons


and polyhedra. The initiator and generator of most of the fractals that we will

discuss will be line segments. The generator is constructed from the initiator,

and then the fractal is constructed from the generator in accordance with a

particular iterative process. After each successive iteration, the shape loses its

traditional Euclidean appearance and, in the limit, becomes a fractal set.

Definition 3.1.1 (Prefractal). If a set X of points is the nth level approxi-

mation of a fractal F , then X is called a prefractal. In particular, X is the

prefractal Fn.

Example 3.1.1 In Figure 3.1, we have the fourth level approximation of a

fractal tree.

Figure 3.1: Fourth level approximation of a fractal tree.

The prefractal in Figure 3.1 is not very interesting, but it illustrates the use

of an initiator and generator to construct a fractal shape. In our example, the

initiator is a line segment and the the generator, a ∨-shape, gives rise to a tree

that, in the limit, will not resemble any Euclidean shape. The importance of

fractal trees will become evident when we begin discussing Fractal Billiards in

Section 10.7 of Chapter 10.

3.2. Recursion and Fractal Construction 31

3.2 Recursion and Fractal Construction

Definition 3.2.1 (Recursion). The process of defining an object in terms of

itself.

Definition 3.2.2 (Algorithm). An algorithm is a fnite set of precise instruc-

tions for performing a computation, for describing a set, or for solving a prob-

lem.

In our discussion, we will mostly be using the word algorithm in the context

of describing a set. From these two definitions, it is clear what a recursively

defined algorithm is.

Definition 3.2.3 (Recursively Defined Algorithm). A finite set of instructions

for which the output of the algorithm is used as input for the same algorithm.

For many fractals, there are many options for symbolically representing the

algorithmic construction. In particular, we will focus on the use of L-Systems

for describing the construction of a variety of fractal sets.

L-Systems, named after the botanist Aristid Lindenmayer, is a formal lan-

guage used to describe the formation of plants. In his research, Lindenmayer

defined variables and operations that served as input for a computer, which

would then output a fairly detailed rendition of the plant. Abstractly, the con-

struction of any object which exhibits self-similarity or self-affineness can be

described by a particular L-system, meaning that every self-similar or self-affine

fractal can be described by a particular L-system. However, computational

restraints restrict an L-system from realistically describing a fractal F .


ANGLE An angle θ measured in degrees.+ Rotation of ANGLE in the counter-clockwise direction.- Rotation of ANGLE in the clockwise direction.

LENGTH The length of the line described by F.SCALE The value by which each segment F is scaled.F Draw a line of a predetermined length in the direction of ANGLE.X No action required. Mainly used as a place holder in complicated L-systems.L Move forward in the direction of ANGLE without drawing a line.

Table 3.1: L-System commands and syntax.

ANGLE 60LENGTH 1SCALE 1/3

INITIATOR -90F

GENERATOR F+F--F+F

Table 3.2: L-system for the Koch curve.

For the purposes of our discussion of L-Systems, we shall use a simplified

version of the language developed by Lindenmayer, which is listed in Table

3.2. In particular, we shall present simple algorithms in a pseudocode format.

In other words, with a mild amount of effort, one could translate a particular

L-system into a working computer program.

Example 3.2.1 The Koch curve K, as seen in Figure 3.2, is constructed by

means of a recursive algorithm. Hence, there is an L-system that describes K

in terms of its generator and initiator, which is summarized in Table 3.2

For the purposes of clarity, one can describe the construction of the Koch

curve in more words. Begin with the initiator, an interval [a, b], and divide it

into three equal thirds. Then, erect an equilateral triangle with sides having

length equal to 13 (b − a) on the middle third. Delete the base of this new


INTEGER i Keep track of the recursion depth.INTEGER r The desired recursion depth.CHAR STRING SHAPE The string that will contain the data.LOOP i Start the looping process.if i = 0 then Decide the depth of recursion.Set SHAPE to INITIATOR

else Depth of recursion greater than 0.Set each F in SHAPE to GENERATOR

INCREMENT i Increase i by 1.END LOOP WHEN i = r

Table 3.3: Pseudocode implementation of the Koch curve L-system.

triangle, and then divide each resulting segment into three equal parts. Again,

construct an equilateral triangle at the middle third of each segment with sides

having length equal to 13 (b′ − a′), where b′ and a′ represent the endpoints of

each resulting segment. In Figure 3.2, we see how the generator is constructed

from the initiator and how each successive prefractal Kn approaches the Koch

curve. After infinitely many iterations of the recursively defined algorithm, the

Koch curve is the resultant figure.

Figure 3.2: The Koch curve. Construction as put forth by von Koch.

Example 3.2.2 The Sierpinski gasket, shown in Figure 3.3, satisfies the previ-


ously discussed criteria, as listed below.

• The Sierpinski gasket is self-similar, since it is the result of the union of a

finite number of similarity transformations.

• The generator of the Sierpinski gasket is an equilateral triangle.

• The Sierpinski gasket can be constructed by way of a recursive algorithm,

as listed in Table 3.2.


INITIATOR -90F+F+F

GENERATOR F+F-F-F+F

Table 3.4: L-system for the Sierpinski gasket.

Example 3.2.3 The Cantor set is the most important in the collection of math-

ematical monsters. Its lack of application and inability to be clearly illustrated

made it most attractive to pure mathematicians. It was a freak of mathematical

nature because of its early inability to be amenable to Mother Nature. As time

progressed, the Cantor set became more and more useful, especially in the study

of chaotic dynamical systems.

The usual construction of the Cantor set is the construction by tremas. The

word trema is used to denote a hole or cut and originates from the Latin word

for termites, tremes. The Cantor set is another creature with fractal properties.

The cardinality of the Cantor set is infinity, but the Cantor set has an infinite

number of holes. It is constructed by:


Figure 3.3: The first five iterations in the construction of the Sierpinski gasket.

1. Dividing the unit interval [0, 1] into three equal lengths.

2. Removing the segment ( 13 , 2

3 ).

3. Dividing the two remaining intervals [0, 13 ] and [23 , 1] into three equal parts

and repeat.

In keeping with the theme of the chapter, we will refer to the Cantor set

L-system in Table 3.2 when discussing the construction. Notice that there are

two generators for the Cantor set L-system.

The Cantor set L-system gives rise to the graphical representation of the

Cantor set in Figure 3.4.



INITIATOR -90F

GENERATOR F FLF

GENERATOR L LLL

Table 3.5: L-system for the Cantor set.

Figure 3.4: Construction of the Cantor set

3.3 Self-Similar and Self-Affinity

At one point in life, everyone has witnessed what happens when two mirrors are

placed opposite of each other.

As Figure 3.5 indicates, at each level, one sees a reflection of a reflection, of

a reflection, and so on. Furthermore, each reflected image can be scaled by a

particular constant a so that it resembles the original set of reflected images,

as the illustration indicates. In order to undertand this phenomenon in full

generality, we will first develop many elementary concepts.1

Definition 3.3.1. τ(x) is a linear translation if,

τ(x) = x + c, (3.1)

1Though such concepts will be defined for vetctors in the plane in terms of complex num-bers, there is no reasy why they following concepts cannot be extended to higher dimensions.Such cases will be examined in the later chapters

3.3. Self-Similar and Self-Affinity 37

Figure 3.5: Reflection of light between two mirrors as an example of self-similarity.

where x is a vector in C.

Definition 3.3.2. r(x) denotes a rotation of the point x ∈ C, if,

r(x) = xeiθ, (3.2)

where θ is the angle of rotation.


Figure 3.6: Linear Translation.

Figure 3.7: Rotation through an angle θ.

Definition 3.3.3. λ(x) denotes a dilation, meaning that every point x ∈ X ⊆ C

is scaled by a particular factor a ∈ R if,

λ(x) = ax. (3.3)

A similarity transformation, denoted by σ(x) is a transformation that pre-

serves angle. Specifically, a similiarity transformation is any combination of


Figure 3.8: Scaling a square by 12 .

translations, rotations or dilations, i.e.,

σ(x) = r ◦ λ ◦ τ(x) (3.4)

= a(x + c)eiθ, (3.5)

where we recall that the symbol ◦ denotes composition of functions.

A self-similarity transformation of a set X is built out of X and finitely

many similarity transformations. If we denote by F the union of the images of

X under a finitely many similarity transformations σ1, σ2, ..., σn, and F is set-

equivalent to the set X, then we call the set X self-similar under the similarity

transformations σ1, σ2, ..., σn, with the caveat that the scaling ratio 0 < a < 1.

Hence X = F or equivalently,


X = σ1(F ) ∪ σ2(F ) ∪ ... ∪ σn(F ) (3.6)

More to the point, any set constructed by means of a self-similarity transforma-

tion is self-similar.

Unfortunately, as a tool for discerning a shape’s fractality, the definition of

self-similarity is too restrictive. Hence, we must develop a more general concept

of self-similarity that includes a larger class of sets. This concept is called self-

affinity.

An affine transformation is a similarity transformation with the added qual-

ity of shearing.

Definition 3.3.4 (Shear – Horizontal Line Preserving). Every point (x0, y0) in

the plane is translated along a vertical line y = y0 that is parallel to a particular

fixed line in the plane, i.e.,

s[(x, y)] = (tx, y) + c (3.7)

where t ∈ R

Definition 3.3.5 (Shear – Vertical Line Preserving). Every point (x0, y0) in

the plane is translated along a vertical line x = x0 that is parallel to a particular

fixed line in the plane, i.e.,


s[(x, y)] = (x, uy) + c (3.8)

where u ∈ R

Definition 3.3.6 (Shear). Every point (x, y) in the plane is translated along a

line that is parallel to a particular fixed line in the plane, i.e.,

s[(x, y)] = (tx, uy) + c (3.9)

where t, u ∈ R and 0 < t < 1, 0 < u < 1.

For example, one can rotate, translate and scale a square, but shearing will

cause the square to become a rhombus. Thus, affine transformations do not

necessarily preserve angle. The following is the form of an affine tranformation

in full generality.

α((x, y)) = r ◦ s((x, y)) (3.10)

= [(tx, uy) + c]eiθ (3.11)

Much like the concept of self-similarity, a self-affine transformation of a set

X is a specific type of affine transformation. If we denote the union of a finite

number of affine transformation α1, α2, ..., αn by F and F is set-equivalent to the


set X, then we refer to the set X as self-affine under the affine transformations

α1, α2, ..., αn, with the caveat that the scaling ratio 0 < a < 1.2 More to the

point, any set constructed by means of a self-affine transformation is self-affine.

Figure 3.9: An affine transformation will not necessarily preserve angle.

From this, we can see that a self-similarity transformation is always a self-

affine transformation, but a self-affine transformation is not necessarily a self-

similarity transformation. In fact, if t = u, then α[(x, y)] is a similarity transfor-

mation. Thus, it may be more natural to describe an affinity transformation as

a similarity transformation with n-many (not necessarily distinct) scaling ratios.

Note that this is not to be confused with the previous footnoted fact tthat each

similarity transformation in the union F may have a unique scaling ratio.

Example 3.3.1 In looking at the Koch curve in Figure 3.10, we see that each

covered region is similar to the whole.

Figure 3.10 is indicative of the fact that there are particular similarity trans-

formations that produce the the Koch curve. They are as follows.

2There is nothing prohibitting each αi from having a unique scaling ratio. Such a scenariowill be discussed in later chapters.


Figure 3.10: Illustrating the result of the similarity transformations on KC.

σ1(x) =1

3x (3.12)

σ2(x) =1

3e

13 πix (3.13)

σ3(x) =1

3e

23 πix +

2

3(3.14)

σ4(x) =1

3x +

2

3(3.15)

Line 3.12 in the list of similarity transformations merely scales every point

of the Koch curve by 1/3. Line 3.13 is the composition of a rotation r(x) = e13 πi

and a scaling λ(x) = 13x. Line 3.14 is the composition of a rotation, scaling, and

translation and Line 3.15 is the composition of a scaling transformation and a

linear transformation. Essentially, Lines 3.14 and 3.15 are the mirror images of

Lines 3.12 3.13.

Example 3.3.2 An example of a self-similar set is the pentagasket. The Eculideian

generator of the pentagasket is a regular pentagon. Applying a particular self-


similarity transformation, we see that we end up with the following fractal shape.

For clarity, we have demonstrated the first four iterations of the recursive algo-

rithm that gives rise to the pentagasket. Notice that if we were to continue the

construction, the nth level approximation of the pentagasket would be nearly

indistinguishable from the (n + 1)th level approximation.

Figure 3.11: The first four iterations in the construction of the pentagasket.

From this, one can see that it is possible to scale, rotate and translate five

copies of the pentagasket by 0 < a < 1 and recreate the original pentagasket.

Example 3.3.3 The Cantor set.

3.4. Infinite Length and Uncountability 45

σ1(x) =x

3(3.16)

σ2(x) =x + 2

3(3.17)

3.4 Infinite Length and Uncountability

Definition 3.4.1 (Finite Set). If a set X has finitely many elements, then we

refer to the set as finite.

Definition 3.4.2 (Countably Infinite). If the elements of a set X can be put

into a bijective correspondence with the natural numbers, then we refer to the

set as countably infinite.

As will be shown later, in order to show something is not countable, it is

enough to show that any attempt to ‘list’ the elements results in at least one

element being unlistable.

Definition 3.4.3 (Uncountable). If a set X cannot be placed into a bijective

correspondence with the natural numbers and the set is not finite, then we refer

to the set X as uncountable.

As sets, most fractals are uncountable, and as geometric objects, it is usually

the case that a fractal will have infinite length. To demonstrate these qualities,

we will prove that the Cantor set is uncountable, and we will attempt to calculate

the length of the Sierpinski gasket and the Koch curve.


Example 3.4.1 Since the Sierpinski gasket is constructed by means of an L-

system, we can view it as a curve and each iteration as a collection of line

segments in the plane. The first iteration of the Sierpinski gasket L-system

shows that the length of the curve is 9/2. The second iteration shows that the

length of SG2 is 27/4. From this, we would like to deduce that the length of

the Sierpinski gasket is 3(

32

)n.

Theorem 3.4.1 (Length of the prefractal SGn). If SGn is the nth level approx-

imation of the Sierpinski gasket, then the length of SGn is given by l(SGn) =

3(3/2)n.

Proof We shall proceed by induction. The trivial case is n = 1. The

prefractal SG1 does indeed have length 9/2. Suppose the length of SGn is

given by l(KCn) = 3(3/2)n for some fixed n. Then,

l(SGn+1) = l

(3

2SGn

)

=3

2l(SGn)

= 33

2

(3

2

)n

= 3

(3

2

)n+1

�

From this, we can see that as n → ∞, l(SGn) → ∞.

Example 3.4.2 By now, one should be familiar with the construction of the

3.4. Infinite Length and Uncountability 47

Koch curve. After the first iteration of the Koch curve L-system, there are four

segments each with length 1/3. After the second iteration, there are sixteen

segments of length 1/9. What we need to first prove is that the length of any

given prefractal KCn can be given by (4/3)n.

Theorem 3.4.2 (Length of the prefractal KCn). If KCn is the nth level ap-

proximation of the Koch curve, then the length of KCn is given by (4/3)n.

Proof We shall proceed by induction. The trivial case is n = 1. The

prefractal KC1 does indeed have length 4/3. Suppose the length of KCn is

given by (4/3)n for some fixed n. Then,

l(KCn+1) = l

(4

3KCn

)

=4

3l(KCn)

=4

3

(4

3

)n

=

(4

3

)n+1

�

From this, we can see that as n → ∞, l(KCn) → ∞.

Example 3.4.3 At first glance, Figure 3.4 would lead one to believe that C is

finite and not terribly large. However, the case is quite the contrary. Mandelbrot

referred to the Cantor set as Cantor dust because the visual representation leads

the reader to believe it is nothing more than specs of dirt laid out in a line. In


fact, the Cantor set is an uncountable subset of the unit interval [1, 0] and has

measure zero, which will be proved later.

It is true that every element of the Cantor set can be described by a ternary

expansion. In other words, each element x of the Cantor set can be written

down as an infinite sequence of zero’s and two’s, i.e.

0.0220200002222...

However, it is not true that the Cantor set is countable, for which a proof

follows.

Theorem 3.4.3. The Cantor set is uncountable.

Proof We shall proceed by finding a contradiction by using the famous

Cantor Diagonalization Method. Suppose the Cantor set is countable. Then

examine only the decimal portion of the ternary representation of x and then

represent the two’s in each sequence with one’s, without changing the nature of

the sequence. Next, attempt to list each sequence in the Cantor set by arranging

each in a table-like format:

1000000000...

0100000000...

0101000101...

0100011101...

...

3.5. Non-Integer Dimension 49

Define the sequence C = {c1, c2, c3, ... : ci = 0 if ci,i = 1 or ci = 1 if ci,i = 0},

where ci,j is the diagonal element of each row in the table above. However,

this sequence, given its construction, will not appear in the listing above, since

every element in the sequence differs from every diagonal element of every listed

sequence. Hence the Cantor set is uncountable.

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3.5 Non-Integer Dimension

The dimension of the Sierpinski gasket is not an integer. It is approximately

D = 1.58 meaning that the Sierpinski gasket has infinite length and zero area,

which is in accordance with the previously mentioned idea that a shape with a

dimension between 1 and 2 would exhibit such behavior.

The last concept in the list of fractal criteria is extremely crucial for under-

standing the very nature of the discipline. In reading the last criterion, certain

questions may arise. First, what is a non-integer dimension. Second what does

it tell us about a fractal F that an integer dimension cannot? In particular,

what do integer dimensions describe about traditional Euclidean shapes.

For the rest of our discussion of non-integer dimensions, allow length, area,

and volume to be denoted by the general term measure. In general, an n-

dimensional Euclidean shape will have a finite measure. A 1-dimensional object

has finite length and zero area, a 2-dimension object has finite area and zero

volume. Therefore, the dimension of a set indicates which measure will be finite


and which measure will be zero.

A natural question is how does one describe the measure of a shape with a

non-integer dimension? Will it indicate a finite measure, and is that measure

length, area, volume or something entirely different? Another question one may

ask is can the measure be denoted as length for a dimension d that approaches

1 and as area for d that approaches 2? Furthermore, will different methods for

calculating the dimension yield different answers? If so, can logical bridges be

made between such methods?

All of the questions, and more, that you may have are valid and have been

addressed by many researchers in the past and present. Hence, answers to these

questions will range from simplistic to very technical.

First, let us answer what happens when a set has a non-integer dimension

d between 1 and 2. Intuitively, one would expect that the set F would not

have a measure bigger than a 2-dimensional object and not smaller than a 1-

dimensional object. The 1-dimensional measure of a d-dimensional object will

be infinity and the 2-dimensional measure will be zero, or finite (but usually

zero). However, it is not always the case that the d-dimensional measure of a

d-dimensional set will be finite. Such a situation will be discussed in more detail

later when we develop the necessary theorems and definitions from Measure and

Dimension theory.

State the self-similarity dimensions of the Cantor set and Koch Curve with-

out proof and show hwo these dimensions indicate infinite zero-dimensional

measures and 0 1 dimesnional measure and infinite 1 dimensional measur eand

3.6. Various Fractal Dimensions 51

zero 2-dim measure, respectively.

3.6 Various Fractal Dimensions

Now give definitions/derivations of particular dimension concepts, i.e. self-

similarity dimension, Hausdorff, box counting, compass dim, etc...point out

benefits, drawbacks, use thesis.

3.7 Summary

Criteria 3.7.1 (Fractal Criteria). A set F ⊆ Rn with (a majority of, if not all

of) the following properties is considered a fractal:

• Usual Euclidean shapes may be used in the construction, but the set F is in

no way easily described by traditional polygons. Furthermore, describing

the local geometry is equally awkward.

• The set F is constructed by means of a recursive algorithm.

• F is self-similar

• The set F has a fine structure, which means that the structure of the fractal

becomes increasingly detailed as the scale m increases.

• Construction of the visual representation of the set F is fairly simple.

• The set F is uncountably infinite and does not usually have an integer

dimension, meaning that F ’s length, area or volume may be infinite or

zero.

An Invitation to Fractal Geometry and Its Applications

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