Theoretical and Experimental

Algorithms Concerning the Dimensions of Fractals

In the

Secondary Classroom

Joyce Eveland

Iowa State University

MSM Creative Component

Summer 2006

Heather Thompson, Major Professor

Leslie Hogben, Co-Major Professor

Alejandro Andreotti, Committee Member

Section 1

Introduction to Fractals and

Present Day Applications

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“From 1975, when Mandelbrot coined the word fractal, to 1980, the word

appeared in just a handful of academic papers. By 1990, there were 5,000 papers a year

published with the word ‘fractal’ in the title.” [12, p. 168] A recent Google search for

fractal returned approximately 276,000 images and 16,100,000 hits in less than a second.

There are hundreds of sites devoted to teaching fractal geometry at all levels of

educational development.

Never before have mathematical insights – usually seen as dry and

dusty – found such rapid acceptance and generated so much excitement in

the public mind. Fractals and chaos have literally captured the attention,

enthusiasm and interest of a world-wide public. To the casual observer,

the color of their essential structures and their beauty and geometric form

captivate the visual senses as few other things they have ever experienced

in mathematics. To the student, they bring mathematics out of the realm of

ancient history into the twenty-first century. And to the scientist, fractals

and chaos offer a rich environment for exploring and modeling the

complexity of nature. [16, p. vii]

Fractal art is found everywhere: on calendars, Web sites, posters, advertisements,

and prints of landscapes that people often mistake for the real thing. Computer programs

allow one to create beautiful fractal screen savers and wall paper. Web sites have fractal

music to play with, http://www.reglos.de/musinum. The mathematics of fractals has a

depth that encourages ongoing research but also many elementary properties that are

accessible to a K-12 audience.

Simmt & Davis provide a fractal card activity in Mathematics Teacher [19, p. 102]

as a way to introduce students to fractals. The repetition of measuring, folding and

cutting gives a tactile understanding of iteration and self-similarity. The conflicting idea

of ever increasing perimeter and ever decreasing area is modeled with finesse. The

creation of a product motivates students and immerses them in the investigation. Brent

Davis used this activity during an introductory, methods for teaching mathematics course.

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His reflections show how this activity motivates students and the diverse nature of

mathematics represented.

Within a few minutes, all present had generated a card, and a good

portion of the class had begun experimenting with other cuts, folds, and

combinations. Most of the introductory three-hour time block was devoted

to playing with the activity: exploring variations, attempting to recreate

completed cards, noticing relationships, generalizing patterns, and

identifying topics in mandated curriculum documents that might be

addressed through this activity. As it turns out, virtually every topic in our

provincial curriculum was at least touched on. [4, p. 29]

Fractals appear and are applied in the study of many areas of science and

medicine. In geology fractals are used to model the earth’s seismic activity to help predict

earthquakes. Spring wire manufacturers examine the fractal nature of the molecular

structure of the wire to determine if it will make a spring. They are able to save time and

money with the more efficient fractal test. Cell phone antenna with fractal structure

provides the best reception. Hydrologists use historical rainfall patterns to predict the

possibility of major rainfall that could precipitate a flood. Architects that design stadiums

and arenas use Orchid fractals to simulate crowd flow. Some economists are trying to use

fractal geometry to study the stock market in hopes of making better predictions when

major trends will occur. The military uses fractal knowledge to locate man-made

structures in natural environment, such as the wake from a submarine. [12] Two recent

applications come from the divergent areas of medicine and art.

In the medical field fractals are used to help document the progression of

Parkinson’s disease. Researchers and doctors in Japan and the United States have

developed a portable system to measure the gait of people with Parkinson’s disease. The

measurements of their movement in three dimensions together with the patients’ walking

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speed are entered into a computer and analyzed using a fractal system. A smooth, steady

walk is comparable to a line and therefore has a dimension of one. In Euclidean geometry

a line has dimension one, a plane dimension two, and a cube dimension three. In fractal

geometry a dimension can be other than a whole number. The researchers found that

healthy elderly subjects have a fractal measure of 1.3 while patients with Parkinson’s

disease have a measure of 1.48 or higher. Doctors hope to use the fractal measure as a

guide to the progression of the disease. Caregivers may not notice a small change in gait

that the computer analysis would find. The change could be an indication that medication

changes are needed. This is important because some of the leading drugs used to help

patients have severe side effects, so getting the right dosage is a quality of life issue. The

body also builds up a tolerance to the drugs so there is a need to continually assess and

adjust the prescriptions. [17, p. 8]

Fractal geometry is also used to document the authenticity of art. Fractal

geometry is the geometry of the irregular shapes we find in nature – which will be

discussed in greater detail in Section 2. In art circles millions of dollars are at stake

concerning thirty-two newly found works of art reportedly by Jackson Pollock. [1]

Richard Taylor, a University of Oregon physicist with an art degree, first used fractal

geometry to analyze Pollock paintings in the late 1990s. By using the box-counting

method – explained in more depth in section two – Taylor claims Pollock’s paintings

have fractal dimensions from close to 1.0 to 1.72. The later works contain the greater

dimension. [22] Using the same techniques Taylor examined the newly found works of

art and reached the conclusion that Pollock may not have painted them. Ellen Landau,

professor of art history at the Cleveland Museum of Art/Case Western Reserve

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University disagrees with Taylor’s analysis. Professor Landau asserts that Pollock’s later

works were influenced by the works of Herbert Matter. The art world has accepted the

opinion of Professor Landau as several of the paintings are being sold as having been

painted by Pollock.

As a branch of mathematics fractal geometry is very young. From 1875 to 1925

some fractals were designed to contradict the prevailing mathematics. Everyone viewed

the fractals as monsters with little value for the physicist trying to explain Nature. Hardly

any new monsters were created for fifty years. Benoit B. Mandelbrot brought the

monsters out of obscurity when during his research he found that fractals could “serve as

the central conceptual tool to answer some old questions that Man had been asking about

the shape of his world.” [13, p. C16] Mandelbrot credits the acceptance of fractal

geometry to computer graphics but gives computers a peripheral role in its genesis. While

most applications of fractals require technology, the framework of fractals lies in the

study of abstract space. Maurice Frechet inaugurated the study of abstract space in 1906.

A space became a set of objects, usually called points, together with a set of relations in

which these points are involved. The set of objects may either be connected or

disconnected. The intuitive sense of connection will suffice at this point. A more precise

definition is found on page 17. The set of relations to which the points are subjected is

called the structure of the space.

Fractal geometry is concerned with the structure of subsets of various spaces. It is

the space on which fractals are drawn; the place where fractals live. These spaces

combined with some measurement function have a number of general properties in

common. The next section will start with an overview of some of these properties.

Section 2

Mathematics of Fractal Dimension

Theoretical and Experimental

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Two forms of dimension will be studied. To lay the mathematical foundation for

each, the following definitions and proofs are given.

Dimension and Topology

Using topological geometry is one way to assign space a dimensional number.

Topology is informally referred to as rubber sheet geometry. An often used example is a

doughnut and a coffee cup. Imagine a doughnut constructed of moldable clay. By

stretching and shaping without tearing (technically a one-to-one transformation or

homeomorphism) one could morph the doughnut into the coffee cup. The hole in the

doughnut becomes the whole in the handle of the cup. Both objects have a genus of one

since they each have one hole. A genus is a classification of a topological property, in this

case the hole that remained invariant under homeomorphism.

Another way to classify topological sets is by dimension. Before defining

topological-dimension Munkres [15, p. 302] explains what is meant by order and

refinement. Let C be a collection of subsets of space X . C has order 1m + if some point

of X lies in 1m + elements of C , and no point of X lies in more than 1m + elements of C.

Using the same set C a collection B is said to refine C, or be a refinement of C, if each

b B∈ is contained in at least one element of C.

Figure 1

Set B

Set C

DE F

E F D

Figure 1 shows a segment, X, covered by a collection of open intervals, set C. C

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:d X X× → �

has an order of 3 since point E lies in 3 intervals and no other point lies in more than 3,

thus m = 2. Also shown is the same segment X, covered by a refining set B, with an order

of 2 since point E lies in 2 intervals and no other point lies in more than 2 intervals, thus

m = 1. Also, all elements of B are contained in at least one interval of C.

Definition 1: A space X is said to be finite-dimensional if there is some integer m

such that for every open covering C of X, there is an open covering B of X that refines C

and has order at most m + 1. The topological dimension of X is defined to be the smallest

value of m for which this statement holds. [15]

Referring back to Figures 1, the smallest value of m is 1 therefore the line

segment has a topological dimension of 1. A point has dimension 0 and a square has

dimension 2.

Regardless of the type of space or the dimension being explained, to further

describe and understand dimension a mathematical way of measuring the distance

between two entities is needed as is the concept of connectedness.

Metric Space

A metric is a function which provides a way of measuring a space. The metric is

commonly denoted by d. A metric space is a space, say X, combined with a metric: (X,d).

Definition 2: A metric space (X,d) is a space X together with a real-valued

function (where � denotes the real numbers), which measures the

distance between pairs of points x and y in X , requiring that d obeys the following

axioms:

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A space together with a metric is called a metric space. A commonly used space is

the n-dimensional Euclidean space, n� where 1 =� � is the real number line and 2

� is

the Euclidean plane. The distance between two points ( , ) ,d x y x y x y= − ∈� is the

Euclidean metric on� . The Euclidean metric on 2� is 2 2

1 1 2 2( , ) ( ) ( )d x y x y x y= − + − ;

the proof that this is a metric follows.

Axiom 1: ( , ) ( , ) ,d x y d y x x y X= ∀ ∈

Let 1 2 1 2( , ) and ( , )x a a y b b= =

( )

( ) ( )

( )

2 2

1 1 2 2

2 2 2 2

1 1 1 1 2 2 2 2

2 2 2 2

1 1 1 1 2 2 2 2

2 2

1 1 2 2

, ( ) ( )

2 2

2 2

,

d x y a b a b

a a b b a a b b

b b a a b b a a

b a b a

d y x

= − + −

= − + + − +

= − + + − +

= − + −

=

Axiom 2: 0 ( , ) x,y , x yd x y X< < ∞ ∀ ∈ ≠

Let 1 2 1 2( , ) and ( , )x a a y b b= =

Show 2 2

1 1 2 20 ( ) ( )a b a b< − + − < ∞

At least one of the differences is positive and the other is non-negative. The sum

of a positive and a non-negative is positive so the square root of the number must be

positive. The two points x and y are distinct points so the distance cannot be zero. The

distance between the coordinates a1 and b1 1 1a b− is finite, likewise 2 2a b− is finite.

1) ( , ) ( , ) ,

2) 0 ( , ) , ,

3) ( , ) 0

4) ( , ) ( , ) ( , ) , , (Triangle Inequality)

d x y d y x x y X

d x y x y X x y

d x x x X

d x y d x z d z y x y z X

= ∀ ∈

< < ∞ ∀ ∈ ≠

= ∀ ∈

≤ + ∀ ∈

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Axiom 3: ( , ) 0 d x x x X= ∀ ∈

Let 1 2( , )x a a=

2 2

1 1 2 2( ) ( ) 0a a a a− + − =

Axiom 4 : ( , ) ( , ) ( , ) , ,d x y d x z d z y x y z X≤ + ∀ ∈

In order to prove this, the Schwartz inequality is needed. A proof of it follows.

Proof of Schwarz Inequality

First prove 2 22ab a b≤ +

2

2 2

2 2

( ) 0

2 0

Thus, 2

a b

a ab b

ab a b

− ≥

− + ≥

≤ +

2 2 2 2

1 2 1 2

Let For 1,2i ix ya b i

x x y y= = =

+ +

2 2

1 1 1 1

2 2 2 2 2 2 2 2

1 2 1 2 1 2 1 2

2 2

2 2 2 2

2 2 2 2 2 2 2 2

1 2 1 2 1 2 1 2

2

2

x y x y

x x y y x x y y

x y x y

x x y y x x y y

≤ + + + + +

≤ + + + + +

Add the two equations together.

2 2 2 2

1 1 2 2 1 2 1 2

2 2 2 22 2 2 21 2 1 21 2 1 2

2 2 2 2

1 1 2 2 1 2 1 2

2x y x y x x y y

x x y yx x y y

x y x y x x y y

+ + + ≤ + + ++ +

+ ≤ + +

Schwartz inequality proven.

2 2 2 2

1 1 2 2 1 2 1 2x y x y x x y y+ ≤ + +

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Prove: ( , ) ( , ) ( , ) , ,d x y d x z d z y x y z X≤ + ∀ ∈

1 1 1 1 1 1

2 2 2 2 2 2

1 1 1 1

2 2 2 2

Let

Thus

a x z b z y

a x z b z y

a b x y

a b x y

= − = −

= − = −

+ = −

+ = −

Show

( ) ( )

( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

( )

2 2 2 2 2 2

1 1 2 2 1 2 1 2

22 2 2 2

1 1 2 2 1 1 2 2

2 2 2 2

1 1 1 1 2 2 2 2

2 2 2 2

1 2 1 1 2 2 1 2

2 2 2 2 2 2 2 2

1 2 1 2 1 2 1 2

22 2 2 2

1 2 1 2

2

1 1 2

2 2

2

2 (By the Schwartz Inequality)

a b a b a a b b

a b a b a b a b

a a b b a a b b

a a a b a b b b

a a a a b b b b

a a b b

a b a b

+ + + ≤ + + +

+ + + = + + +

= + + + + +

= + + + + +

≤ + + + + + +

= + + +

+ + +( )2 2 2 2 2

2 1 2 1 2a a b b≤ + + +

Definition 3: Two metrics d1 and d2 on a space X are equivalent if there exist

constants 0<c1<c2<∞ such that 1 1 2 2 1( , ) ( , ) ( , ), ( , )c d x y d x y c d x y x y X X≤ ≤ ∀ ∈ × [2]

Any pair of equivalent metrics has the same notion of which points are close

together and which are far apart. After distances are determined between points the set is

deformed in a standard way- i.e. stretched or contracted with a transformation function –

with no tears, rips, overlaps or infinite compression or stretching. Then the distances are

determined again. (Figure 2)

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A pair of points x and y in .

Let the Euclidean distance between these points

be d1(x,y) . A function f transforms these points

and the new Euclidean distance is called d2(x,y).

The two metrics may be equivalent if the

deformation leads to no rips, tears, overlapping

or infinite stretching.

Definition 4: Two metric spaces 1 1 2 2( , ) and ( , )X d X d are equivalent if there is a

function 1 2:h X X→ that is one-to-one and onto (i.e. it is invertible), such that the

metric �1d on 1X defined by � ( )1 2 1( , ) ( ), ( ) , ,d x y d h x h y x y X= ∀ ∈ is equivalent to d1.

Properties of Subsets of Metric Space

Completeness

Subsets of metric spaces (X,d) are described, classified, analyzed, and observed in

fractal geometry. The properties of openness, closedness, boundedness, completeness,

and compactness – the definitions are developed in the following pages – are invariant

under equivalent metric spaces. The importance of this is the relationship between

subsets of equivalent spaces. As an example, let space X containing subset A, be

deformed with bounded distortion and name the new space Y where Y contains subset B

which is the image of A. If subset A has one of the aforementioned properties then subset

B will also have that property.

Fractals live in complete metric spaces so that is a good property to start with. The

definition of completeness depends upon understanding the Cauchy sequence and limits.

A definition for a Cauchy sequence follows.

Figure 2

Y

X

Y

X

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Definition 5: A sequence 1{ }n nx∞

= of points in a metric space (X,d) is called a

Cauchy sequence if, for any given number 0ε > , there is an integer 0N > so that

( , )n md x x ε< for all ,n m N> . [2]

A sequence { }nx whose elements get arbitrarily close together for sufficiently

large n and m is a Cauchy sequence. This does not guarantee that the sequence is

converging toward one point, leading to the next definition.

Definition 6: A sequence 1{ }n nx∞

= of points in a metric space (X,d) is said to

converge to a point x X∈ if, for any given number 0ε > , there is an integer 0N > so

that ( , ) for all nd x x n Nε< > . In this case the point x X∈ , to which the sequence

converges, is called the limit of the sequence, and we use the notation lim nn

x x→∞

=

Theorem 1: If a sequence of points{ }1n n

x∞

=in a metric space (X,d) converges to

a point x X∈ , then { }1n n

x∞

=is a Cauchy sequence. [2]

Proof: Let { }nx x→ in space X and 2 0ε > . There exists an N such that

( , ) 2 if nd x x n Nε< > . For n,m N> ( , ) ( , ) ( , )n m n md x x d x x d x x ε≤ + < by the triangle

inequality. Thus it is a Cauchy sequence.

Definition 7: A metric space (X,d) is complete if every Cauchy sequence 1{ }n nx∞

= in

X has a limit x X∈ . [2]

Closure

Definition 8: Let S X⊂ be a subset of a metric space ( , )X d . The closure of S,

denoted, S is defined to be S S= ∪ {Limit points of S in X}. S is closed if it contains all

of its limit points in X, that is, S S= . [2]

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Open

Definition 9: Let S X⊂ be a subset of a metric space (X,d). S is open if for each

x S∈ there is an ε > 0 such that ( , ) { : ( , ) }B x y X d x y Sε ε= ∈ < ⊂ . [2]

Note that by definition X is closed though X may not be complete. Let

( , ) { : ( , ) }B x y X d x yε ε= ∈ ≤ denote a closed ball of radius 0ε > centered at x. A closed

ball contains its bounding sphere and an open ball ( , ) { : ( , ) }IntB x y X d x yε ε= ∈ < does

not. The limit x of a convergent sequence 1{ }n nx∞

= has this property: any such ball centered

at x contains all of the points xn after some index N, where N typically becomes larger and

larger as ε becomes smaller and smaller. In 2� the ball is a disc described with a center

point and a radius. In 1� a ball is an interval that can be open (0,1) , closed [0,1], and

half-open(0,1], or half-closed [0,1).

Compactness

Definition 10: A metric space X is compact if every open covering has a finite

subcovering. [18]

That is, there is a finite collection { }1 2

1

such that N

n i

i

O ,O ,...,O X O .µ=

⊂ =∪

Definition 11: Let S X⊂ be a subset of a metric space ( , )X d . S is compact if

every covering µ of S by open sets of X has a finite subcovering.

If sequence n

x x→ then it can be said that each ball about x contains infinitely

many elements of the sequence. Therefore, x is a cluster point of the sequence.

Definition 12: Given 0ε > and given N, there is an n N≥ so that n

d( x,x ) ε<

then x is a cluster point of xn.

Thus if x is a limit of xn then x is a cluster point. The converse of this is not true.

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A space X has the Bolzano-Weierstrass property if every infinite sequence xn in X

has at least one cluster point. Meaning if there is an x X∈ where each open set

containing x and for each N there is an with n

n N x O.≥ ∈

Theorem 2: Let X be a metric space then: X is compact, X has the Bolzano-

Weierstrass property, and X is sequentially compact are equivalent. [18, p. 155]

Bounded

Definition 13: Let S X⊂ be a subset of a metric space ( , )X d . S is bounded if

there is a point a X∈ and a number R>0 so that ( , ) d a x R x S< ∀ ∈ .

Definition 14: Let S X⊂ be a subset of a metric space ( , )X d . S is totally bounded

if, for each 0ε > , there is a finite set of points 1, 2{ ,..., }ny y y S⊂ such that whenever

, ( , )ix X d x y ε∈ < for some 1 2{ , ,... }i ny y y y∈ . This set of points 1 2{ , ,... }ny y y is called an

ε -net.

Example:

The real numbers are bounded but not totally bounded under the

metric ( ) { }3d x,y min ,x y= − .

Figure 3

10 -5

Boundedbecause alldistanceswill becontained bya ball ofradius 3.

Real Numbers

d(x,y)=min{3, x-y }

0

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Figure 4

-5 5 10

Not totallybounded, let εεεε = 1then there is not afinite number ofballs - contained inthe εεεε-ball - that willcover the set.

0

Compact ↔Closed and Totally Bounded

Theorem 3: Let (X,d) be a complete metric space. Let S X⊂ . Then S is compact

if and only if it is closed and totally bounded.

Proof: Suppose that S is closed and totally bounded. Let { }ix S∈ be an infinite

sequence of points in S. Since S is totally bounded there is a finite collection of closed

balls of radius 1 such that S is contained in the union of these balls. By Dirichlet’s Box

Principle one of the balls, say B1, contains infinitely many of the points xn. Choose N1 so

that1 1Nx B∈ . It is easy to see that 1B S∩ is totally bounded. So we can cover 1B S∩ by a

finite set of balls of radius1/ 2 . By Dirichlet’s Box Principle, one of the balls, say B2,

contains infinitely many of the points xn. Choose N2 so that 2 2Nx B∈ and 2 1N N> . We

continue in this fashion to construct a nested sequence of balls,

1 2 3 4 ... ...nB B B B B⊃ ⊃ ⊃ ⊃ ⊃ ⊃

Where Bn has radius 1

1

2n−and a sequence of integers 1{ }n nN

∞

= such thatnN nx B∈ . It is easy

to see that 1{ }nN nx

∞= , which is a subsequence of the original sequence {xn}, is a Cauchy

sequence in S. Since S is closed, and X is a complete metric space S is complete as well,

1{ }nN nx

∞= converges to a point in S. Thus, S is compact.

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Conversely, suppose that S is compact. Let 0ε > . Suppose that there does not

exist an netε − for S. Then there is an infinite sequence of points { }nx S∈ with

( , )i jd x x ε≥ for all i j≠ . But this sequence must possess a convergent subsequence

{ }iNx By Theorem 1 this sequence is a Cauchy sequence, and so there is a pair of integers

Ni and Nj with i jN N≠ such that ( , )i jN Nd x x ε< . However, ( , )

i jN Nd x x ε≥ , so we have a

contradiction. Thus an netε − does exist. [2]

The function f(x) approaches a limit L as x approaches some number c when for

any positive numberε , there is a positive number δ such that ( )f x L ε− < if

0 x c δ< − < . The function f is continuous at x = c if and only if lim ( ) ( )x c

f x f c→

= .

Meaning that:

1. f(c) must exist, c is in the domain of f

2. The limit, lim ( )x c

f x L→

= must exist and f(c) must equal L.

Connectedness

Definition 15: A metric space (X,d) is connected if the only two subsets of X that

are simultaneously open and closed are X and ∅ . A subset S X⊂ is connected if the

metric space (S,d) is connected. S is disconnected if it is not connected. S is totally

disconnected provided that the only nonempty connected subsets of S are subsets

consisting of single points. [2]

The closed interval [0,2] is connected and can be written as [ ) [ ]0,1 1,2∪ with the

first interval half-open and the second closed. The union of [0,1) and (1,2] is not

connected because the intervals are simultaneously open and closed. Hence they are

disconnected, but not totally disconnected.

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Definition 16: Let ⊂S X be a subset of a metric space (X,d). Then S is pathwise-

connected if, for each pair of points x and y in S, there is a continuous function

f:[0,1]→S, from the metric space ([0,1],Euclidean) into the metric space (S,d) such that

f(0)=x and f(1)=y. Such a function f is called a path from x to y in S. S is pathwise-

disconnected if it is not pathwise-connected. [2]

Theorem 4: Let X be a path connected space, then X is connected. [14, p. 135]

The groundwork has been laid for the understanding of a complete metric space

such as 2( , )Euclidean� . We will use drawings, “black-on-white” subsets of space to

illustrate the following concepts.

(H(X),h)

Definition 17: Let (X,d) be a complete metric space, then H ( X ) denotes the

space whose points are the compact subsets of X, other than the empty set. [2]

A compact non-empty subset of X is a single point in ( X )Η . Each element of the

space ( X )Η is a compact non-empty subset of X. Let the points in the space 2( )Η � be

demonstrated as black and white images.

The horse’s front half is a point in H(X). Call it ( )x H X∈ .

The back half is a point in H(X). Call it ( )y H X∈

This Black Beauty is x y∪ and forms a new single point in

( X )Η . Unions of points yield new points. [2]

Figure 5

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B

(εεεε,0)

d

c

(x3,y3)

(x2,y2)

ba

(x1,y1)

Definition 18: Let (X,d) be a complete metric space, x X∈ , and B ( X )Η∈ ,

define ( , ) min{ ( , ) : }.d x B d x y y B= ∈= ∈= ∈= ∈ d(x,B) is called the distance from the point x to the

set B. [2]

Example: Calculate d(c,B) if (X,d)

is the space ( )2 2, Euclidean , c ∈� � is the

point (x1,y1) and B is a closed disk of

radius ε centered at the point (ε,0). Let a

and b, b ≠ a be points on the boundary of

the closed disk as indicated in Figure 6.

Let point a be the intersection of the line

connecting points c and d, with the

boundary of the disk (Figure 6). By the

application of Euclidean geometry triangle properties it is clear that ( , ) ( , )d b c d a c≥ .

Clearly the distance from c to any point contained in B and not on the boundary will also

be greater than the distance from c to a.

Definition 19: Let (X,d) be a complete metric space. Let , ( )A B X∈ Η . Define

( , ) max{ ( , ) : }d A B d x B x A= ∈= ∈= ∈= ∈ . d(A,B)is called the

distance from the set ( )A X∈ Η to the set ( )B X∈ Η . Since

A, B are compact there are points ˆ ˆ and x A y B∈ ∈ such that

ˆ ˆ( , ) ( , )d A B d x y= .

Let , ( )A B H X∈ , where ( , )X d is a metric space. In

Figure 7

3

2

1

-1

-2

-3

-2 2

A

B

y

x

Figure 6

20 of 57

general ( , ) ( , )d A B d B A≠ thus showing a lack of symmetry leading to the conclusion that

d does not provide a metric on H(X). Figure 4 shows an example of this. Set A is a disk

with radius 2 and set B is a disk with a radius of 1 such

that B A⊂ , x A∈ and y B∈ .Using ( , ) max{ ( , ) : }d A B d x B x A= ∈ it is shown

that ( , ) 1d A B ≥ . ( , ) max{ ( , ) : }d B A d y A y B= ∈ since B A⊂ then y A∈ and the distance

from a point to itself is zero then ( , ) 0d B A = ; thus showing that ( , ) ( , )d A B d B A≠ .

Definition 20: Let (X,d) be a complete metric space. Then the Hausdorff distance

between points A and B in H(X) is defined by ( , ) ( , ) ( , ) h A B d A B d B A= ∨ using the

notation x y∨ to mean the maximum of the two real numbers x and y.

Show h is a metric on the space H(X).

, , ( )A B C H X∈

{ }( , ) ( , ) ( , ) ( , ) max ( , ) : 0h A A d A A d A A d A A d x A x A= ∨ = = ∈ =

( , ) ( , ) ( , )

( , ) ( , ) ( , )

( , ) ( , )

h A B d A B d B A

h B A d B A d A B

h A B h B A

= ∨

= ∨

=

( , ) ( , ) for some and bh A B d a b a A B= ∈ ∈ using the compactness of A and B. Hence,

0 ( , )h A B≤ < ∞ . If then we can assume there exists an so that ≠ ∈ ∉A B a A a B (or

so that ∈ ∉b B b A ). It follows that 0h( A,B ) d( A,B ) d( a,B )≥ ≥ ≥ .

To show ( , ) ( , ) ( , )h A B h A C h C B≤ + first show that ( , ) ( , ) ( , )d A B d A C d C B≤ + .

21 of 57

{ }

{ }

{ }

{ } { }{ }

Choose any

( , ) min ( , ) :

min ( , ) ( , ) :

( , ) min ( , ) : , so

( , ) min ( , ) : max min ( , ) : :

( , ) ( , ), so

( , ) ( , ) ( , ).

a A

d a B d a b b B

d a c d c b b B c C

d a c d c b b B c C

d a B d a c c C d c b b B c C

d a C d C B

d A B d A C d C B

∈

= ∈

≤ + ∈ ∀ ∈

≤ + ∈ ∀ ∈

≤ ∈ + ∈ ∈

≤ +

≤ +

Similarly,

( , ) ( , ) ( , ), thus

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( , )

d B A d B C d C A

h A B d A B d B A d A C d C A d B C d C B

h A C h C B

≤ +

= ∨ ≤ ∨ + ∨

= +

Theorem 5: The Completeness of the Space of Fractals:

Let (X,d) be a complete metric space. Then (H(X),h) is a complete metric space.

If ( ){ }1n n

A H X∞

=∈ is a Cauchy sequence, then ( )lim n

nA A H X

→∞= ∈ can be characterized

as follows: { }{ }n: there is a Cauchy sequence x that converges to nA x X A x= ∈ ∈ . [2]

The space H(X) and the metric h make up the complete metric space (H(X),h) in

which fractals reside. Knowing where they live prepares us to discuss what they are.

Classic Fractals

Fractals can be very

complex looking but they are

usually constructed with fairly

simple rules. These rules are

Original Line Segment

Step One - Remove the middle 1/3.

Step Two - Remove the middle 1/3 of each segment.

Step Three - Keep removing the middle 1/3 of each segment.

Figure 8

22 of 57

repetitive in nature. The Cantor set shown in Figure 8 is created by removing the middle

one-third of each segment in the previous generation to create the current generation. It is

referred to as the Cantor middle third set. There are different styles of Cantor sets

depending upon how much is removed and what shape the originating figure was. The

Cantor middle fifth set has the middle 1/5 of the set removed.

The Koch curve requires the removal and replacement of segments. This

picture (Figure 9) shows how the middle third of the

initiator is removed and replaced by an equilateral

triangle, without its base. This process is repeated on each

segment of the generator to construct level 2. At stage n

the length is 4 3n n Self-similarity is built into the

process. Each of the four parts in the kth

step is a scaled

down version – by a factor of three – of the entire curve

in the previous (k-1)st step. [16]

Sierpiński’s Triangle, (Figure 10) is another classic fractal built with a simple

recursive rule. By joining the midpoints of each side of an equilateral triangle four

smaller equilateral triangles are formed. Once the center triangle is removed, three

triangles remain and the process is repeated. If the sides of the originating triangle in

Stage 1 have a length of unit one, the sides of the three remaining triangles in Stage 2

each have a length of ½ unit. In Stage 3 nine triangles have sides of length ¼ unit.

Therefore Stage k will have 3k-1

triangles each with a side length of 1

1

2k−unit.

Figure 9

23 of 57

Properties of Fractals

Falconer presents a concept of fractal based upon five properties (F is a subset of

a metric space):

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

2. F is too irregular to be described in traditional geometrical language, both

locally and globally.

3. Often F has some form of self-similarity, perhaps approximate or statistical.

4. Usually, the ‘fractal dimension’ of F (defined in some way) is greater than its

topological dimension.

5. In most cases of interest F is defined in a very simple way, perhaps

recursively.

The first property refers to the fact that one can zoom in on areas of a fractal and

see a picture analogous to the original. This is easily seen with the Koch curve – also

known as von Koch curve, coastline, or snowflake (Figure 11) depending upon the

completed shape. The Sierpiński’s triangle also displays the fine structure as seen in

Figure 12.

Figure 11

The second property requires that set F be too irregular to be described in

Figure 12 – Sierpinski Triangle Area to zoom shown in box

Area in box magnified

Koch curve under magnification.

24 of 57

traditional geometrical language, both locally and globally. This is demonstrated in the

afore mentioned Koch curve. A traditional curve is globally smooth and locally has the

property of tangency. The Koch curve is very rough and nowhere differentiable. Jean

Perrin described this irregularity in a philosophical manifesto in 1906 – presented in free

translation by Mandelbrot. [13, p. 7]

Consider, for instance, one of the white flakes that are obtained by

salting a solution of soap. At a distance its contour may appear sharply

defined, but as we draw nearer its sharpness disappears. The eye can no

longer draw a tangent at any point. A line that at first sight would seem to

be satisfactory appears on close scrutiny to be perpendicular or oblique.

The use of a magnifying glass or microscope leaves us just as uncertain,

for fresh irregularities appear every time we increase the magnification,

and we never succeed in getting a sharp, smooth impression, as given, for

example, by a steel ball. So, if we accept the latter as illustrating the

classical form of continuity, our flake could just as logically suggest the

more general notion of a continuous function without a derivative.

Property three which states, F often has some form of self-similarity, perhaps

approximate or statistical, allows us to define sets as fractals even if they do not have

complete exact self-similarity. The Cantor set (Figure 8) has fine structure and excellent

self-similarity. While a cauliflower (see Figure 13) does not posses fine structure or

excellent self-similarity it is often used as an example of a natural object having some

fractal like properties. Each small piece approximately looks like the whole but is not an

exact copy. This is why a cauliflower is considered to have approximate self-similarity.

Figure 13

25 of 57

The fourth property regards the dimension of fractals, particularly the relationship

of the topological dimension and the Hausdorff dimension. An informal look at

Hausdorff dimension considers the number of balls – N(r) – of radius r required to cover

A completely. As r decreases in size N(r) increases. If N(r) increases at the same rate as

1 dr as 0r → then A has Hausdorff dimension of d. A formal definition occurs later in

this section. The fourth property is revisited on page 38.

The simple recursive rules for creating the Cantor set, Koch’s curve and

Sierpiński’s triangle exemplify the last property on Falconer’s list. The act of repeatedly

removing a triangle results in the paradox of increasing perimeter and decreasing area, a

complex idea modeled by simplistic means.

Dimension

The foundation has been laid to begin the discussion of dimension in relation to

fractals. In comparing Koch’s snowflake with the Cantor set, is one larger than the other?

How can they be measured? Fractal dimension is the common way to compare fractal

sets. Dimension is a subjective feeling about how densely an object occupies the space in

which it lies. Real world fractal-like entities such as: coastlines, clouds, trees, feathers,

networks of neurons in the body, dust in the air at an instant of time, the waves and

ripples of the sea, and the distribution of frequencies of light reflected by a flower can

have their fractal dimension measured. The most common hands-on method consists of

counting the number of boxes intersected by an image of the entity covered by grids of

various sizes. An example of this is found on page 35. When a dimension is found for the

object it can then be compared to the dimension of mathematically created fractals, like

those classic fractals previously mentioned. One thing both methods have in common is

26 of 57

the restriction to compact subsets of metric spaces.

N(A, ε)

Let (X,d) denote a complete metric space. Let ( )A H X∈ be a nonempty compact

subset of X. Let 0ε > . Let ( , )B x ε denote the closed ball of radius ε and center at a

point x X∈ . Let the integer ( , )N A ε be the least number of closed balls of radius ε

needed to cover set A.

1( , ) smallest positive integer such that ( , ) for some set of

distinct points { : 1,2,..., }

M

n n

n

N A M A B x

x n M X

ε ε== ⊂ ∪

= ⊂

Using an open ball with radius 0ε > to surround every point x A∈ provides a

covering of set A by open sets. A is compact so by definition it possesses a finite

subcover, integer M̂ , of open balls. By taking the closure of each ball there now exists a

cover of M̂ closed balls. Let C denote the set of covers of A by at most M̂ closed balls of

radius ε, then C is not empty. Thus it contains at least one element. Let

ˆ: {1,2,..., }f C M→ be defined by ( ) number of balls in thef c = cover c C∈ . Then

{ ( ) : }f c c C∈ is a finite set of positive integers. Thus it contains a least integer ( , )N A ε .

[2]

Fractal Dimension

Set A has fractal dimension D if:

( , ) for some positive constant C.DN A Cε ε −≈ *

Solving for D, ln ( , ) ln

ln(1/ )

N A CD

ε

ε

−≈

* For real valued function of positive real value ε ( ) ( )f gε ε≈ means { }

0

lim ln ( ) ln ( ) 1f gε

ε ε→

= .

27 of 57

The notation ln(x) denotes the logarithm of the base e of the positive real number

x. The term ( )1ln ln approaches zero as 0C εε

→ , leading to the following definition.

Definition 21: Let ( )A H X∈ where (X,d) is a metric space. For each ε>0 let

( , )N A ε denote the smallest number of closed balls of radius ε>0 needed to cover A. If

( )( )0

ln ( , )lim

ln 1/

N AD

ε

ε

ε→

=

exists, then D is called the fractal dimension of A. The notation

D=D(A) is read A has fractal dimension D. [2]

For example, let (X,d) be a metric space. Let , , and let { , , }a b c X A a b c∈ = .

We can show that D(A)=0.

Let (X,d) be a metric space, with { , , }.A X A a b c⊂ = Calculate the fractal

dimension of A. Let ,min ( ( , ))x y Ar d x y∈= . For any then ( , ) 32

rN Aε ε< = .

Thus the fractal dimension is0

ln 3( ) lim 0

1ln

D Aε

ε

→= =

.

Fractal Dimension – Square and Koch Curve

To show that a square has fractal dimension of 2, let the unit square A be in the

metric space ( )2 , Euclidean� . Let ε > 0. The square can be covered by 2( 1)2 n− closed

disks of radius ( ) ( )1 2 2n

for all 1n ≥ .

( )0

ln( ( , ))lim

ln 1

N AD

ε

ε

ε→

=

Let 2 2nε = for 1n ≥ . As , 0n ε→ ∞ → and 2( 1)( , ) 2 nN A ε −= .

28 of 57

( )2( 1)ln 2( ) lim

1ln2 2

2( 1) ln 2lim

ln 2 ln 2

2

n

n

n

n

D A

n

n

−

→∞

→∞

=

− =

−

=

To find the fractal dimension of the Koch curve start with 1 disk with a radius of

1/2 to cover it all. A pattern of 4 disks with a radius of 1/6, 16 disks with a radius of 1/18

develops showing that, in general, it takes 4n disks of radius 1/(2·3

n) to cover the Koch

curve. Let the Koch curve reside in a metric space (X,d) with ( )1 2 3nε = ⋅ for 0n ≥ . As

, 0n ε→ ∞ → and ( , ) 4nN A ε = .

( )

( )

ln 4( ) lim

1ln

12 3

ln 4lim

ln 2 ln 3

ln 41.261859507...

ln 3

n

n

n

n

D A

n

n

→∞

→∞

= ⋅

=

+

= ≈

The variable ε is continuous in Definition 21, but we have been using ε in a

discrete manner in the examples. The following theorems will justify this use.

Theorem 6: Let ( )A H X∈ , where (X,d) is a metric space. Let n

n Crε = for some

real numbers 0 1 r< < and 0C > , and integers 1,2,3,...n = . If

29 of 57

( )ln( ( , ))

lim1ln

n

Fn

n

N AD

ε

ε→∞

=

,then A has fractal dimension DF.

Proof: Let the real numbers r and C, and the sequence of numbers

{ : 1,2,3,...}nE nε= = be as defined in the statement of the theorem. Let

( ) max{ : }n nf Eε ε ε ε= ∈ ≤ with rε ≤ , then ( ) ( ) / andf f rε ε ε≤ ≤

( ) ( ) , ( ) ( , ) , ( ) /N A f N A N A f rε ε ε≥ ≥ .

Since ln(x) is an increasing positive function of x for 1x ≥ ,

( )( )( )

( ) ( )( )( )

ln , ( ) / ln , ( )ln ( , )

ln 1/ ( ) ln(1/ ) ln / ( )

Expression 1 Expression 2 Expression 3

N A f r N A fN A

f r f

ε εε

ε ε ε

≤ ≤

Assume that ( , ) as 0N A ε ε→ ∞ → ; if this is false then the theorem must be true.

Expression 3:

( )( )( )

( )( )( )

( )( )

( )( )

0

ln , ( ) ln ,lim lim

ln / ( ) ln /

ln ,lim

ln( ) ln(1/ )

ln ,lim

ln(1/ )

n

nn

n

nn

n

nn

N A f N A

r f r

N A

r

N A

ε

ε ε

ε ε

ε

ε

ε

ε

→ →∞

→∞

→∞

=

=+

=

Expression 1:

( )( )( )

( )( )( )

( )( )( ) ( )

( )( )( )

1

0

1

1

ln , ( ) / ln ,lim lim

ln 1/ ( ) ln 1/

ln ,lim

ln 1/ ln 1/

ln ,lim

ln 1/

n

nn

n

nn

n

nn

N A f r N A

f

N A

r

N A

ε

ε ε

ε ε

ε

ε

ε

ε

−

→ →∞

−

→∞−

→∞

=

=

+

=

Expressions 1 and 3 are approaching the same value when ε →0. By the

30 of 57

squeeze law of calculus the limit as 0ε → of Expression 2 also exists, and it equals the

same value. [2]

Box-counting Dimension

One of the most widely used dimensions is the Box-counting or box dimension.

This method goes back at least until the 1930s and has been known by a variety of

names: Kolmogorov entropy, capacity dimension, metric dimension, logarithmic density,

and information dimension. Its popularity stems from the relative ease of mathematical

computation by both man and machine. Plus, it can be used on any structure in a plane

and be adapted for structures in space. [7] An informal explanation is to place a grid of

uniform boxes – having side length s – over the structure and count the number of boxes

that contain some of the structure. Let this number be N. The number of boxes depends

upon the size of the grid so let this relationship be represented by Ns. Repeat with a grid

of smaller size. The goal is to use several grids with s getting smaller each time ( 0s → ).

A stat-plot of ( )log / log(1/ )sN s will linearize the data so a trend line can be predicted

and its slope measured. The slope is the box-counting dimension, DB, of the structure.

Theorem 7: The Box Counting Theorem

Let ( )mA H∈ � , where the Euclidean metric is used. Cover m� by closed square

boxes of side length (1/2n). Let Nn(A) denote the number of boxes of side length (1/2

n)

which intersect the set. If ln( ( ))

limln(2 )

nB nn

N AD

→∞

=

,then A has fractal dimension DB.

Proof: For ( )1 ( )

11,2,3,..., ( ,1/ 2 ) for all 1,2,3,...,

2 1

n

n k nmm N N A N n−= ≤ ≤ =

+where

k(n) is the smallest integer k satisfying 2

11 log

2k n m≥ − + . The first inequality holds

31 of 57

2

1

21

Figure 15

because the number of boxes of size 11

2n− needed to cover the structure is less than or

equal to the number of boxes of size 11

2n− needed to cover a ball cover of A where each

ball has radius of 12n . Since each ball of radius 1

2n can intersect at most 2m +1 on-

grid boxes of side length 11 2n− , the number of boxes needed to cover the specified ball

cover of A is less than or equal to (2m+1)·N(A,1/2

n).

For example let m = 2. In 2� a unit square with n = 2, a ball of

radius 1/4 will intersect at most 5 on-grid boxes of side length 1/2. (Figure

15)

The second inequality follows from the fact that a box of side s

can fit inside a ball of radius r provided

2 2 2 2

2 ...2 2 2 2

s s s sr m

≥ + + + =

by the

theorem of Pythagoras. Using s� as the highest power of 1/2 that is less than s, let

1 and

2k( n )s s s≤ =� � then

1

1 1

2 2k( n ) n

m−

≤ . Solving for k(n)

1

2 2 2

2

2

2 2

1 2 2

1

11

2

n k( n )m

( n ) log log m k( n )log

( n ) log m k( n )

( n ) log m k( n )

− ≤

− ∗ + ≤

− ∗ ≤

− ∗ ≤

We want the least k that makes this true.

The ( )

( )( )( )

( )( )

( )

( ) ( )

( )

ln 2ln lnlim lim

ln 2 ln 2 ln 2

k n

k n k n

n n k nn n

N ND

→∞ →∞

= =

since

( )1.

k n

n→

32 of 57

Since also 11

1

1

1ln

ln2lim limln(2 ) ln(2 )

nmn

n nn n

NN

D−+

−

−→∞ →∞

= =

Theorem 6 with 1 2r = completes the proof. [2, with additions and corrections]

If the box-counting technique is being used on a natural object, generally an image of

the object in 2� will be covered with a grid of squares. By using the image of the object only

the dimension of the image will be found, not the actual object. This is just one of the

drawbacks to the box-counting algorithm. Another is that the boxes are counted as either

containing or not containing a part of the object. No weight is given to how much of the object

appears in the box. A box containing one point counts the same as a box filled less one point.

This leads to the geometrical structure of the fractal set being analyzed while ignoring the

underlying measure. There are numerous other dimensions and various algorithms to find them

that remove these limitations. Other techniques allow any metric space to be used. Then one

could use balls to measure the distance between points on a fractal, possibly giving a more

accurate answer. It must be remembered that any dimension found with a numerical technique

is an estimate only. The box-counting technique is well suited to the secondary curriculum.

The mathematics is attainable yet not typical, helping it to be refreshing.

33 of 57

1 11 2, ( ) 2s N s= =

Here are examples illustrating the box-counting technique.

Let s be the size of the box and N(s) be the number of boxes needed.

( ) ( )

( ) ( )

( ) ( )

(1) 1

1 2 2 1/ 1 2

1 4 4 1/ 1 4

and so in general

1/

N

N

N

N s s

=

= =

= =

=

( ) ( )

( ) ( )

log / log 1

log 4 / log 1/ (1 4) 1

N s

=

This relationship holds for every size box so the

dimension of a line segment is 1, as was

expected.

Unit Square

s0=1, N(s0)=1

s1=1/2, N(s1)=4

s2= 1/4, N(s2)=16

Box SizeNumber

of Boxes

s N(s) Log 1/ s Log N(s)

1 1 0.0000 0.0000

1/2 4 0.3010 0.6021

1/4 16 0.6021 1.2041

y = 2x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

The slope of the line is 2 as expected.

Line

segment

0 01, ( ) 1s N s= =

2 21 4, ( ) 4s N s= =

N(1)=1

( )

( )

1 2 4

log 42

log1/ 1 2

N =

=

( )

( )( )

1 4 16

log162

log 1/ 1 4

N =

=

34 of 57

Using the Box-Counting Technique on Natural Fractals

In nature there are no true fractals but there are objects that possess fractal

behavior. Tree branching, clouds formations, and lightning are just a few. Using the box-

counting technique a photo of lightning is analyzed for possible fractal dimension.

Begin by making the assumption that the

original photograph had a dimension of 1unit

by 1 unit. Then each of the squares would

have a length of 1/2 unit. 4 out of 4 squares

contain part of the lightning.

Each square is now 1/4 the length of the original

square. 14 out of 16 contain lightning.

Figure 16 Figure 17

35 of 57

Figure 18

40 out of 64 squares contain lightning and the

sides of the squares are now 31 2 1 8= the size

of the original.

Each square is 1/16 the size of the

original and 99 of the squares

contain part of the lightning.

Figure 19

36 of 57

Each square is 1/32 the size of the

original and 233 of the squares

contain part of the lightning.

Hausdorff-Besicovitch Dimension

The Hausdorff–Besicovitch dimension of bounded subsets of m� is another way

to describe the dimension of a set. It is associated with a method that can be used to

compare the sizes of subsets with equal fractal dimensions. The mathematics used is

more complex and does not lend itself to experimental procedures to determine the fractal

dimension of physical sets.

The definition will be restricted to metric space ( ),m d� where m is a positive

integer and d denotes the Euclidean metric. Let mA ⊂ � be bounded. Then the diameter of

Size of

Box

No. of

Boxes

Log (1 /

Box Size)

Log(Box

Count)

0.03125 233 1.51 2.37

0.0625 99 1.20 2.00

0.125 40 0.90 1.60

0.25 14 0.60 1.15

0.5 4 0.30 0.60

By this analysis it appears the lightning

has a box-counting dimension of 1.455.

Log-log Graph

y = 1.455x + 0.2286

0.00

1.00

2.00

3.00

0.00 0.50 1.00 1.50 2.00

Figure 20

37 of 57

A is { }( ) sup ( , ) : ,diam A d x y x y A= ∈ . The supremum, least upper bound, might not be

attained by any elements ,x y A∈ .

Let 0 and 0 .pε< < ∞ ≤ < ∞ Let A denote the set of sequences of subsets

{ } 1, such that i i iA A A A∞

=⊂ = ∪ . Then define

{ }1

( , , ) inf ( ( )) : , and ( ) for 1,2,3,... .p

i i i

i

M A p diam A A A diam A iε ε∞

=

= ∈ < =

∑

Here we use 0( ( )) 0 when i idiam A A= is empty. ( , , )M A p ε is a number in the range [ ]0,∞ ;

its value may be zero, finite, or infinite. Now define { }( , ) sup ( , , ) : 0M A p M A p ε ε= >

Then for each [ ] [ ]0, there is ( , ) 0,p M A p∈ ∞ ∈ ∞ . [2]

Definition 22: Let M be a positive integer and let A be a bounded subset of the

metric space ( , )m Euclidean� . For each [ )0,p ε∈ the quantity ( , )M A p described above

is called the Hausdorff p-dimensional measure of A. [2, p. 198]

The Hausdorff p-dimensional measure ( , )M A p , as a function of [ ]0,p ∈ ∞ , has a

range consisting of only one, two, or three values. The possible values are zero, a finite

number, and infinity. This leads to the following theorem.

Theorem 8: Let m be a positive integer and let A be a bounded subset of the metric

space ( ),m Euclidean� . Let ( , )M A p denote the function of [ )0,p ε∈ defined above.

Then there is a unique real number [ ]0,HD m∈ such that

[ )

[ )

if and 0,( , )

0 if and 0,

H

H

p D pM A p

p D p

ε

ε

∞ < ∈ =

> ∈

[2]

Definition 23: Let m be a positive integer and let A be a bounded subset of the

38 of 57

metric space ( ),m Euclidean� . The corresponding real number HD , occurring in

Theorem 8 is called the Hausdorff-Besicovitch dimension of the set A. This number will

also be denoted by ( )HD A .

Theorem 9: Let m be a positive integer and let A be a subset of the metric space

( ),m Euclidean� . Let ( )D A denote the fractal dimension of A and let ( )HD A denote the

Hausdorff-Besicovitch dimension of A. Then ( ) ( )0 .HD A D A m≤ ≤ ≤ [2]

In 1918 Felix Hausdorff introduced the Hausdorff dimension as an extended non-

negative real number associated to any metric space. Abram Samoilovitch Besicovitch

developed methods of computing the Hausdorff dimension of highly irregular sets.

Today, the Hausdorff-Besicovitch dimension is generally referred to as simply the

Hausdorff dimension. Sometimes it is also called the capacity dimension or fractal

dimension.

Returning to Falconer’s fourth property of fractals (page 25), Mandelbrot first

defined a fractal as “a set for which the Hausdorff-Besicovitch dimension – DH – strictly

exceeds the topological dimension.”[3, p. 15] Mandelbrot recognized that this definition

is exclusive resulting in borderline cases. He chose to keep it

restrictive but encouraged others to make a case for expanding it.

Falconer chooses to expand the definition by adding the word usually

to the front. This allows fractals like the Devil’s staircase, Figure 22 to

be called a fractal even though, DH = 1 = DT. There are many types of staircases but the

most common is based upon the Cantor function. More can be found about this function

at http://mathworld.wolfram.com/CantorFunction.html.

Figure 21

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The three classic fractals described earlier and their associated dimensions are

listed in the following chart. It appears that rounding gives rise to any difference between

the Hausdorff and box-counting dimensions.

Summary

Fractal geometry is the relation between mathematical sets and natural objects.

There are no true fractals in the physical world but if viewed under certain ranges of scale

some objects will exhibit fractal behavior. The experimentalist must decide what scale

makes the most sense to use for each application. An interesting example of this is a ball

of 10 cm diameter made of a thick thread of 1 mm diameter exhibits several distinct

dimensions.

To an observer placed far away, the ball appears as a zero-

dimensional figure: a point. … As seen from a distance of 10 cm

resolution, the ball of thread is a three-dimensional figure. At 10 mm, it is

a mess of one-dimensional threads. At 0.1 mm, each thread becomes a

column and the whole becomes a three-dimensional figure again. At 0.01

mm, each column dissolves into fibers, and the ball again becomes one-

dimensional, and so on, with the dimension crossing over repeatedly from

one value to another. [13, p. 17]

The film clip of this transition from dimension to dimension captures the fluidness

of so called natural fractals. Perhaps this feeling is why some people find fractal

geometry so fascinating.

Name Topological

Dimension Hausdorff Dimension Box-counting Dimension

Middle Third

Cantor 0 0.63 0.62989

Koch Curve 1 1.26 1.26186 Sierpiński’s

Triangle 1 1.585 1.58996

[16]

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[1] “Art Expert Argues 32 Disputed Works Done by Pollock.” Des Moines Sunday

Register. Feb. 26, 2006, 11E

[2] Barnsley, Michael F. Fractals Everywhere. 2nd

ed. Academic Press

Professional, Cambridge, 1988

[3] Choate, Jonathan, Robert L. Devaney, and Alice Foster. Fractals A Tool Kit of

Dynamics Activities. Key Curriculum Press, Emeryville, CA, 1999

[4] Davis, Brent. “Basic Irony: Examining the Foundations of School Mathematics

with Preservice Teachers.” Journal of Mathematics Teacher Education

(1999): 25-48. Retrieved 6/1/06 from

http://dx.doi.org/10.1023/A:1009942822958

[5] Elert Glen. The Chaos Hypertextbook. 2005. Hypertext.com. 5/25/06. Retrieved

6/21/06 from http://hypertextbook.com/chaos

[6] Eves, Howard. Foundations and Fundamental Concepts of Mathematics. Dover

Publishing, Mineola, 1990: 212-235

[7] Falconer, Kenneth. Fractal Geometry Mathematical Foundations and

Applications. 2nd

ed. John Wiley & Sons Ltd., West Sussex, 2003

[8] Frame, Michael, Benoit Mandelbrot and Nial Neger. Fractal Geometry. Yale

University. Retrieved 6/4/06 from http://classes..yale.edu/Fractals/

[9] “Limit of a Sequence.” Linear Mathematics in Infinite Dimensions Signals

Boundary Value Problems and Special Functions. May 2006. Ohio State

University. Retrieved 6/5/06 from http://www.math.ohio-

state.edu/~gerlach/math/Bvtypset/node9.html

41 of 57

[10] “Cauchy Sequence.” Linear Mathematics in Infinite Dimensions Signals

Boundary Value Problems and Special Functions. May 2006. Ohio State

University. Retrieved 05/22/06 from http://www.math.ohio-

state.edu/~gerlach/math/Bvtypset/node10.html

[11] Hoggard, John. “Fractal Geometry.” 5/2/97. Virginia Tech. Retrieved 6/25/06

from http://www.math.vt.edu/people/hoggard/FracGeom

Report/FracGeomReport.html

[12] Lesmoir-Gordon, Nigel, Will Rood, Ralph Edney. Introducing Fractal

Geometry. Totem Books, Lanham, 2001

[13] Mandelbrot, Benoit. The Fractal Geometry of Nature. W. H. Freeman and

Company, New York, 1983

[14] Mendelson, Marion. Introduction to Topology. Mineola: Dover, 1990

[15] Munkres, James. Topology A First Course. Prentice Hall: Englewood, NJ,

1975: 300-305

[16] Peitgen, Heinz-Otto, Hartmut Jurgens, and Dietmar Saupe. Chaos and Fractals

New Fronteirs of Science. New York: Springer-Verlag New York 1992

[17] “Physicists Use Fractals To Help Parkinson’s Sufferers.” Institute of Physics 4

Feb. 2004. Retrieved 6/10/06 from

http://www.sciencedaily.com/releases/2004/02/040203232954.html

[18] Royden, H. L. Real Analysis. 3rd

ed. Mac Millian Publishing Co, New York,

1988: 146-155

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[19] Simmt, Elaine and Brent Davis. “Fractal Cards: A Space for Exploration in

Geometry and Discrete Mathematics.” Mathematics Teacher 91 (Feb.

1998): 102-8

[20] Strogatz, Steven H. Nonlinear Dynamics and Chaos. Westview Press, 1994:

398-416

[21] Taylor, Richard P. “Order in Pollock’s Chaos.” Scientific American Dec. 2002:

116-21

[22] Taylor, Richard P., Adam P. Micolich and David Jonas. “Fractal Analysis of

Pollock’s Drip Paintings.” June 3, 1999. Retrieved 06/23/06 from

http://www.uoregon.edu/~msiuo/taylor/art/Nature/.pdf

[23] Wright, David J. “Fractal Dimension.” Dynamical Systems and Fractals Lecture

Notes. Aug. 19, 1996. Oklahoma State University. Retrieved 5/21/06 from

http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node37.html

Appendix

44 of 57

Constructing Fractals Using

Geometer’s Sketchpad

Objectives: Students will use Internet resources and Sketchpad technology to construct several

classic fractals and design one of their own.

Use this site – http://www.math.wsu.edu/faculty/vincent/INME/chaosfacilitator.htm – to build

Koch’s Curve, Sierpiński’s Triangle, Pythagorean Fractal and your own creation.

After opening a new sketch window go to File → Document Options → Add Page →

Blank. After clicking on Blank go to the space for page name and type in the name of the fractal

you will be building on that page, including your original. Construct and name four pages then

click on OK. You may also rename pages after they have been created by using the document

options menu.

On the web site there is a separate student’s page. Open a word processing document and

the student’s pages link. Use copy and paste to transfer the questions from the web page to your

document. Answer the questions using your document. Make sure to save on a regular basis,

every ten minutes or so.

When your project is completed save the sketch file and the document file to the

class drop folder.

45 of 57

Student handout answer key.

1) Why does the triangle CDE need to be an equilateral triangle?

The equilateral triangle guarantees that the four segments we are wanting to use

are all approximately congruent.

2) What does the iteration tool do to the images? In mathematical terms describe

what happens when you iterate.

The image of segments AC, CE, ED and DB are dilated with a fractional scalar

then merged so that A→A and B→C this same image is then mapped to the

remaining points: A→C and B→E; A→E and B→D; A→D and B→B. Iterate – do a

process over again.

3) Describe the process to draw and iterate Koch’s Curve.

Start with a segment, AB and construct two points randomly between the

endpoints, C and D. Label the points. Select all of the points in order from left to

right. Construct a segment between each pair of points. (Ctrl + L) With the segments

highlighted measure their lengths. Drag the points until the lengths are equal. Select

one of the interior points and the middle segment. Construct a circle with a point and

radius. Choose the other interior point and middle segment and construct a second

circle. Construct the intersections of the two circles. Label the top point E. Select

points C, D and E. Construct segments joining these three points. It should form

equilateral triangle CDE. Hide everything but points A, B, C, D, and E and segments

AC, CE, ED, and DB. Select points A and B. Under the translation menu choose

iterate. Map A→A and B→C, choose Add new mappings under the Structure menu

and continue. A→C and B→E; A→E and B→D; A→D and B→B. On the display

menu choose Final Iteration Only. Click on Iterate.

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4) What does this fractal look like? Describe it in words and try to draw it here.

The fractal looks starry or like the edge of a snowflake. Triangles on top of triangles.

5) What happens when you move point A or

B?

The curve grows in all directions at a steady and equal rate.

6) What happens when you move point C towards A? Towards D? Towards B?

When point C moves toward A the middle of the curve grows taller while

the outer portions shrink. When C moves toward D the triangles get smaller and

the curve smoothes until if C=D the curve becomes a line. Once C moves past D

towards B the triangles grow on the underside of the curve. If C = B then it forms

a triangle similar to Sierpin ski triangle.

7) What happens when you move point E towards A or B? Can you describe the

motion of point E? (Hint: select point E and choose Trace Intersection under the

Display menu.)

Point E traces a path of what appears to be the top and sides of a trapezoid.

As E is moving toward A the triangles on the B side grow and the A side shrink

while the middle grows. The same pattern holds when E is moving toward B.

Sierpiński’s Triangle

Sierpiński’s Triangle begins with a simple triangle. It then constructs the midpoints of

each segment of the triangle and creates another triangle inside of the first by connecting

all the midpoints. It doesn’t look or sound all that interesting at first, but when you apply

the iteration a number of times, what comes out is pretty neat.

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1) What can you tell about the area of the new triangles being created? How does it

relate to the area of the original triangle?

The new triangles are 1/4 the area of the previous triangle. The new triangle will

be 1/4n of the original triangle.

2) If you increase the number of iterations is it possible for there to be an infinite

number of triangles within Sierpiński’s Triangle?

The triangles could go on infinitely but the computer would run out of

memory and the display would not be capable of displaying it.

3) What do you notice about the triangles created by the iteration?

The triangles are becoming very small very quickly and there number is getting

very large very quickly.

4) Would you expect anything about the fractal, other than the general shape, to

change if one of the vertices is changed? Why or why not?

No, the number of triangles would remain the same and the area would

change in proportion to the distortion caused by dragging a vertex.

5) How do the interior triangles relate to the exterior triangles? Is there anything that

can be said about how each iteration relates to the previous iteration?

The interior triangles are always larger than the exterior triangles and the area

appears to be in a ratio of 1/4n where n relates to the number of iteration. Each

triangle turns into four new congruent triangles with every iteration.

The Pythagorean Fractal

The fractal you are constructing comes from the Pythagorean Theorem. By using this

diagram and iterating it we can create an interesting picture.

48 of 57

E

DC

Α

Β

1) Can you think of another way to construct a square using Geometer’s Sketch Pad?

Explain your idea here. Answers may vary.

Create two points and a line segment. Select the two points and segment,

using the transformation menu, translate the image using rectangular and marked

distance: zero for horizontal and marked distance for vertical. By tabbing to the

vertical option and then selecting the segment the computer will figure the

distance. Connect the points with segments.

2) What does the iteration look like? Will the image look different when you change

the number of iterations?

The image looks like a flower. With more iterations

it will have a larger bloom.

3 - Iterations

49 of 57

3) What can be said about the triangles being formed by the iterations?

The triangles are all isosceles right triangles. Every triangle at each stage

gives birth to two new triangles at the next iteration. If you square the length of

one of the legs of a new triangle and multiply that answer times 2 that will equal

the length of the hypotenuse of the originating triangle squared.

4) Are the squares related to each other? If so, how? If not, why not?

The side of the new square is related by the following mathematical

connection. ( )2

2 new side old side= They are also related in that the area of the

two new squares equals the area of the old square.

5) Remember, the Pythagorean Theorem is 2 2 2a b c+ = . How does this fractal relate

to the theorem?

This fractal contains many copies of a geometric proof of the Pythagorean

Theorem.

a b

c

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Dimension Using Similarity

From Euclidean geometry we know that points have zero dimension, a line has one, a

plane two, and space three. This comes from the general idea that points have no length, width, or

depth; lines have one direction, length; planes have two directions, length and width; and space

has three directions, length, width and depth. To discuss dimension using more rigorous

mathematics we must first understand the terms self-similar and magnification.

A straight line has self-similarity because any part of the line looks exactly like the

original, just smaller. Take line segment AB that measures five inches in length. Divide the

segment into five one-inch segments; each segment looks like the original. If we take one of the

segments and magnify it by a factor of five it will be exactly like the original. Notice that the

magnification factor is how many of the new segments it will take to reproduce the original. If the

segment had been divided into 20 self-similar pieces then it would take a magnification factor of

20 to yield the original segment. Therefore, we can break a line into N self-similar pieces with a

magnification factor of N.

A square may be divided into self-similar squares (Figure 1).

A cube may be divided into self-similar cubes (Figure 2).

Letting

Two times the

side of new

square equals

the original.

4 = 22

Three times the

side of new square

equals the

original.

9 = 32

Four times the

side of new

square equals

the original.

16 = 42

For a square the number

of self-similar squares =

(magnification factor)2.

Two times the

side of new

cube equals the

original.

8 = 23

Three times the

side of new cube

equals the

original.

27 = 33

Four times the

side of new

cube equals the

original.

64 = 43

For a cube the number

of self-similar cubes =

(magnification factor)3.

51 of 57

Page 2

Letting m = magnification factor

n = number of self-similar pieces

d = dimension of fractal

then n = md.

For the middle square in figure 1 m = 3 and n = 9.

d

d

d

d

=

=

=

=

2

3log

9log

3log9log

39

Similarly using m = 4 and n = 16 we could show that 2 = d. Thus the dimension of a

square is two.

How about the Sierpin ski Triangle?

It is shown broken into three pieces so

n = _______.

Each would need two more just like it to

complete another Sierpin ski triangle so

m = _______.

Usinglog

thus log

d nn m d

m= =

solve for d, rounding to the nearest

thousandths.

d = ________.*

* If your answer is not between 1 and 2 check your work for errors.

log log

log log

loglog

d

d

n m

n m

n d m

nd

m

=

=

=

=

52 of 57

Page 3

Use the previous method to find the dimension of this Koch curve.

Begin by identifying what a self-similar piece might look like.

Figure 1 Figure 2 Figure 3

m= m= 9 (Hint) m=

n= n= n=

d= d= d=

Which figure do you think

gives the most relevant

answer?

Explain why.

Figure 1 Figure 2 Figure 3

53 of 57

Perimeter of Koch Curve

The perimeter of the Koch curve is ________________________. How many

stages would guarantee a perimeter larger than 100 units ________?

200 units ________?

Area Under Koch Curve

What is meant by area under the Koch curve? This is the area between

stage infinity and stage 0 of the curve. Examine the above picture; stage 0 has

no area and is not represented in the picture, stage 1 has the area of the triangle,

stage 2 has that area plus the 4 new triangles, and stage 3 has the area of stage

2 plus the 16 new triangles. Because we want to look at stage infinity (stage n :

n→∞ ) it appears that new area will always be added. Does this imply that the

Stage Number

of

segments

Length of

each

segment

Total length (in linear units)

0 1 = 1 = 1/30 1 =

1 4 = 1/3 4 · 1/3 = 4/3

2 16 = 42 1/9 = 1/32 16 · =

3

n

Stage3

Stage 2

Stage 1

Stage 0

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area under the Koch curve is infinite? ___________

Support your answer. _______________________________________________

Let’s examine each stage mathematically.

Stage 0 no area

Stage 1 A units2 This area could be computed but if we allow it to be

A then every Koch curve is represented.

Stage 2 4

9A A

+

A is the original triangle plus the 4 new triangles.

Each of the new triangles has a base ______ the size

of the original and a height _______ the size of the

original. Thus, their area is _______ of the original.

Stage 3 _________ Stage 2 + (# new triangles)·(scaled amount of original)·(A)

Rewriting Stage 3 using exponents ______________.

Now write a formula for the area to the nth stage _______________________

and then factor out the A _________________________________________ .

This is a geometric series so using the formula 2 3 11

1r r r ...

r+ + + + =

−we can see

that the area under the Koch curve (where stage 1 has an area of A) is:

2 34 4 4 1 1 9

19 9 9 1 4 9 5 9 5

A ... A A A

+ + + + = = = −

So, what we have just found is the perimeter of the Koch curve is infinite the area

is finite! Next you will find the area and perimeter of the Sierpins

．

ki triangle.

55 of 57

Area of Sierpins

．

ki Triangle

Stage

Number of

triangles removed

Image

Area of each

triangle removed

Total area removed at this stage

Total area removed so

far

0 0

0 0 0

1 1

1/4 1/4 1/4

2 3

1/16 2

13

4

2

1 13

4 4

+

3

4

n

You do not need to create these

images.

The area of the Sierpins

．

ki triangle is ____________because:

______________________________________________________________

______________________________________________________________

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Perimeter Sierpins

．

ki Triangle

When discussing the perimeter of Sierpins

．

ki triangle remember that the

boundary of the removed triangle remains and becomes a part of the original

triangle. Thus, it is included in the perimeter. The following chart will help you

determine the perimeter at each stage and then determine what happens as the

stages increase. Let S be the length of one side of the original triangle.

Stage Number of

sides added

Length of each

added side

Total length added

at this stage

Total length added

so far

0 0 0 0 3S

1 3

2

S 3

2

S

3

32

SS +

2

3

4

n

What is the pattern? What is happening to the perimeter of Sierpinski triangle?

______________________________________________________________

______________________________________________________________

______________________________________________________________

57 of 57

Teacher Notes:

The Koch curve exercises were created to be used as an in-class group

project, probably teacher led. This depends of course on each group of students.

The Sierpinski triangle work was designed for the student to work independently.

Answers:

Perimeter of Koch curve 4

3

n

17 stages gives a perimeter > 100

19 stages gives a perimeter > 200

Area Sierpinski Triangle 0 units – How can it have an area of 0 and still be seen?

A line has an area of 0 and it is still seen, even though it is made up of infinitely

many points.

Perimeter of Sierpinski Triangle is infinite

0

33

2

n

n

S S

∞

=

+