Theoretical and Experimental Algorithms Concerning the Dimensions ... [PDF]

to the front. This allows fractals like the Devil's staircase, Figure 22 to be called a fractal even though, DH = 1 = DT

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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

“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. 2 of 57

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 Frechet 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.

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Section 2 Mathematics of Fractal Dimension Theoretical and Experimental

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

Set B

E

F

D

E

F

D

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

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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 d : X × X →  (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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1) d ( x, y ) = d ( y , x )

∀x , y ∈ X

2) 0 < d ( x, y ) < ∞

∀x , y ∈ X , x ≠ y

3) d ( x, x ) = 0 ∀x ∈ X 4) d ( x, y ) ≤ d ( x, z ) + d ( z, y ) ∀x, y, z ∈ X (Triangle Inequality) 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 d ( x, y ) = ( x1 − y1 ) 2 + ( x2 − y2 ) 2 ; the proof that this is a metric follows. Axiom 1: d ( x, y ) = d ( y , x ) ∀x , y ∈ X

Let x = (a1 , a2 ) and y = (b1 , b2 ) d ( x, y ) = (a1 − b1 ) 2 + (a2 − b2 )2 = a12 − 2a1b1 + b12 + a2 2 − 2a2b2 + b2 2 = b12 − 2b1a1 + a12 + b2 2 − 2b2 a2 + a2 2 2

( b1 − a1 ) + ( b2 − a2 ) = d ( y, x ) =

2

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

Let x = (a1 , a2 ) and y = (b1 , b2 ) Show 0 < ( a1 − b1 ) 2 + (a2 − b2 )2 < ∞

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 a1 − b1 is finite, likewise a2 − b2 is finite.

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Axiom 3: d ( x , x ) = 0 ∀x ∈ X

Let x = (a1 , a2 )

( a1 − a1 ) 2 + ( a2 − a2 ) 2 = 0 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 x1 y1 + x2 y2 ≤ x12 + x22 y12 + y22 First prove 2ab ≤ a 2 + b 2 ( a − b)2 ≥ 0 a 2 − 2ab + b 2 ≥ 0 Thus, 2ab ≤ a 2 + b 2

Let

a=

xi 2 1

2 2

b=

x +x

yi 2 1

y + y22

For i = 1, 2

2

2

2

2

   y1 x1  ≤ 2 2 2  2 2  2 x1 + x2  y1 + y2   x1 + x22

   y1  +    y2 + y2  2    1

   y2 x2  ≤ 2 x12 + x22  y12 + y22   x12 + x22

   y2  +    y2 + y2  2    1

x1

x2

Add the two equations together.

  x2 + x2 y2 + y 2 x1 y1 + x2 y2 2 2 2  ≤ 12 + 12 2 2 2 2 2   x1 + x2 y1 + y22  x1 + x2 y1 + y2  x1 y1 + x2 y2 ≤ x12 + x22 y12 + y22 Schwartz inequality proven.

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

Thus

a1 = x1 − z1

b1 = z1 − y1

a 2 = x2 − z 2

b2 = z2 − y2

a1 + b1 = x1 − y1 a2 + b2 = x2 − y2

Show

( a1 + b1 )

(

2

2

2

+ ( a2 + b2 ) ≤ a12 + a22 + b12 + b22

( a1 + b1 ) + ( a2 + b2 )

2

2

) = (a + b ) + (a + b ) 2

1

1

2

2

2

= a12 + 2a1b1 + b12 + a22 + 2a2b2 + b22 = ( a12 + a22 ) + 2 ( a1b1 + a2b2 ) + ( b12 + b22 ) ≤ ( a12 + a22 ) + 2 =

( a1 + b1 )

2

(

(

)

a12 + a22 b12 + b22 + ( b12 + b22 ) (By the Schwartz Inequality)

a12 + a22 + b12 + b22

)

2

2

+ ( a2 + b2 ) ≤ a12 + a22 + b12 + b22

Definition 3: Two metrics d1 and d2 on a space X are equivalent if there exist constants 0 N . [2] A sequence {xn } 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 {xn }∞n =1 of points in a metric space (X,d) is said to

converge to a point x ∈ X if, for any given number ε > 0 , there is an integer N > 0 so that d ( xn , x ) < ε for all n > N . In this case the point x ∈ X , to which the sequence converges, is called the limit of the sequence, and we use the notation x = lim xn n →∞



Theorem 1: If a sequence of points { xn }n =1 in a metric space (X,d) converges to ∞

a point x ∈ X , then { xn }n =1 is a Cauchy sequence. [2] Proof: Let { xn } → x in space X and ε 2 > 0 . There exists an N such that d ( x, xn ) < ε 2 if n > N . For n,m > N d ( xn , xm ) ≤ d ( xn , x ) + d ( x, xm ) < ε by the triangle

inequality. Thus it is a Cauchy sequence. Definition 7: A metric space (X,d) is complete if every Cauchy sequence {xn }∞n =1 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, ε ) = { y ∈ X : d ( x, y ) < ε } ⊂ S . [2]

Note that by definition X is closed though X may not be complete. Let B ( x, ε ) = { y ∈ X : d ( x, y ) ≤ ε } denote a closed ball of radius ε > 0 centered at x. A closed

ball contains its bounding sphere and an open ball IntB ( x, ε ) = { y ∈ X : d ( x, y ) < ε } does not. The limit x of a convergent sequence {xn }∞n =1 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] N

That is, there is a finite collection {O1 ,O2 ,...,On } ⊂ µ such that X = ∪ Oi . i =1

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 xn → 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 ε > 0 and given N, there is an n ≥ N so that d( x,xn ) < 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 n ≥ N with xn ∈ O.

Theorem 2: Let X be a metric space then: X is compact, X has the BolzanoWeierstrass 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 ε > 0 , there is a finite set of points { y1, y2 ,..., yn } ⊂ S such that whenever x ∈ X , d ( x, yi ) < ε for some yi ∈{ y1 , y2 ,... yn } . This set of points { y1 , y2 ,... yn } is called an

ε -net. Example: The real numbers are bounded but not totally bounded under the metric d ( x,y ) = min {3,x − y} . Figure 3 Real Numbers d(x,y)=min{3, x-y } Bounded because all distances will be contained by a ball of radius 3. 10

-5

0

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Figure 4 Not totally bounded, let ε = 1 then there is not a finite number of balls - contained in the ε-ball - that will cover the set. -5

0

5

10

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 {xi ∈ 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 that x N 1 ∈ B1 . It is easy to see that B1 ∩ S is totally bounded. So we can cover B1 ∩ 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 x N 2 ∈ B2 and N 2 > N1 . We continue in this fashion to construct a nested sequence of balls, B1 ⊃ B2 ⊃ B3 ⊃ B4 ⊃ ... ⊃ Bn ⊃ ... Where Bn has radius

1 and a sequence of integers {N n }n∞=1 such that xN n ∈ Bn . It is easy n−1 2

to see that {xN n }∞n =1 , 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,

{xN n }∞n =1 converges to a point in S. Thus, S is compact.

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Conversely, suppose that S is compact. Let ε > 0 . Suppose that there does not exist an ε − net for S. Then there is an infinite sequence of points {xn ∈ S } with

d ( xi , x j ) ≥ ε for all i ≠ j . But this sequence must possess a convergent subsequence {xN i } By Theorem 1 this sequence is a Cauchy sequence, and so there is a pair of integers

Ni and Nj with N i ≠ N j such that d ( x Ni , x N j ) < ε . However, d ( x Ni , x N j ) ≥ ε , so we have a contradiction. Thus an ε − net does exist. [2] The function f(x) approaches a limit L as x approaches some number c when for any positive number ε , 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 f ( x ) = f (c ) . x→c

Meaning that: 1. f(c) must exist, c is in the domain of f 2. The limit, lim f ( x ) = L must exist and f(c) must equal L. x →c

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. 17 of 57

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 pathwisedisconnected 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 Η ( X ) . Each element of the space Η ( X ) is a compact non-empty subset of X. Let the points in the space Η (  2 ) be demonstrated as black and white images. Figure 5

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

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

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

define d ( x , B ) = min{ d ( x , y) : y ∈ B}. d(x,B) is called the distance from the point x to the set B. [2] Figure 6

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

(x 1 ,y 1 )

is the space (  2 , Euclidean ) , c ∈  2 is the point (x1,y1) and B is a closed disk of radius ε centered at the point (ε,0). Let a

(x 2 ,y 2 ) b

a

(x 3 ,y 3 )

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

the closed disk as indicated in Figure 6.

(ε,0)

Let point a be the intersection of the line connecting points c and d, with the

B

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

Figure 7

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

3

x

2

A, B are compact there are points xˆ ∈ A and yˆ ∈ B such that A

1

d ( A, B ) = d ( xˆ , yˆ ) .

B -2

2

y -1

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

-2

-3

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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 d ( A, B ) = max{d ( x, B ) : x ∈ A} it is shown that d ( A, B ) ≥ 1 . d ( B, A) = max{d ( y , A) : y ∈ B} since B ⊂ A then y ∈ A and the distance from a point to itself is zero then d ( B, A) = 0 ; 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 ) h ( A, A) = d ( A, A) ∨ d ( A, A) = d ( A, A) = max {d ( x, A) : x ∈ A} = 0

h ( A, B ) = d ( A, B ) ∨ d ( B, A) h ( B, A) = d ( B, A) ∨ d ( A, B ) h ( A, B ) = h ( B, A) h ( A, B ) = d (a , b) for some a ∈ A and b ∈ B using the compactness of A and B. Hence, 0 ≤ h ( A, B ) < ∞ . If A ≠ B then we can assume there exists an a ∈ A so that a ∉ B (or b ∈ B so that b ∉ A ). It follows that h( A,B ) ≥ d( A,B ) ≥ d( a,B ) ≥ 0 . To show h ( A, B ) ≤ h( A, C ) + h (C , B ) first show that d ( A, B ) ≤ d ( A, C ) + d (C , B ) .

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Choose any a ∈ A d (a , B ) = min {d (a , b) : b ∈ B} ≤ min {d (a , c) + d (c, b) : b ∈ B}∀c ∈ C ≤ d (a , c) + min {d (c, b) : b ∈ B}∀c ∈ C , so d ( a , B ) ≤ min {d (a , c) : c ∈ C} + max {min {d ( c, b) : b ∈ B} : c ∈ C } ≤ d (a , C ) + d (C , B ), so d ( A, B ) ≤ d ( A, C ) + d (C , B ). Similarly, d ( B, A) ≤ d ( B, C ) + d (C , A), thus 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



{A ∈ H ( X )} n

n =1

is a Cauchy sequence, then A = lim An ∈ H ( X ) can be characterized n →∞

as follows: A = {x ∈ X : there is a Cauchy sequence {x n ∈ An } that converges to 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 Figure 8 Original Line Segment

Fractals can be very

Step One - Remove the middle 1/3.

complex looking but they are

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

usually constructed with fairly

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

simple rules. These rules are

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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

Figure 9

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 n 3n 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 2k −1

unit.

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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 Koch curve under magnification.

Figure 12 – Sierpinski Triangle

Area to zoom shown in box

Area in box magnified

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

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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

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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 r d as r → 0 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

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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 ε > 0 . 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.

N ( A, ε ) = smallest positive integer M such that A ⊂ ∪nM=1 B( xn , ε ) for some set of distinct points {xn : n = 1, 2,..., M } ⊂ X Using an open ball with radius ε > 0 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 Mˆ , of open balls. By taking the closure of each ball there now exists a cover of Mˆ closed balls. Let C denote the set of covers of A by at most Mˆ closed balls of radius ε, then C is not empty. Thus it contains at least one element. Let f : C → {1, 2,..., Mˆ } be defined by f (c ) = number of balls in the 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: N ( A, ε ) ≈ Cε − D for some positive constant C. * Solving for D,

*

D≈

ln N ( A, ε ) − ln C ln(1 / ε )

For real valued function of positive real value ε f (ε ) ≈ g (ε ) means lim {ln f (ε ) ln g (ε )} = 1 . ε →0

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

( ε ) approaches zero as ε → 0 , leading to the following definition.

x. The term ln C ln 1

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  ln ( N ( A, ε ) )  D = lim   exists, then D is called the fractal dimension of A. The notation ε →0  ln (1/ ε )  D=D(A) is read A has fractal dimension D. [2] For example, let (X,d) be a metric space. Let a , b, c ∈ X and let A = {a , b, c} . We can show that D(A)=0. Let (X,d) be a metric space, A ⊂ X with A = {a , b, c}. Calculate the fractal dimension of A. Let r = min x , y∈A ( d ( x, y )) . For any ε <

Thus the fractal dimension is D( A) = lim ε →0

r then N ( A, ε ) = 3 . 2

ln 3 = 0. 1 ln   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 2( n −1) closed disks of radius (1 2 )

n

( 2 ) for all n ≥ 1 .

 ln( N ( A, ε ))  D = lim   ε →0  ln (1 ε )  Let ε = 2 2 n for n ≥ 1 . As n → ∞, ε → 0 and N ( A, ε ) = 2 2( n −1) .

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   ln ( 2 2( n −1) )    D ( A) = lim   n →∞   ln  1  2 2 n      2(n − 1) ln 2  = lim   n →∞ n ln 2 − ln 2   =2 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/(2·3n) to cover the Koch curve. Let the Koch curve reside in a metric space (X,d) with ε = 1 ( 2 ⋅ 3n ) for n ≥ 0 . As n → ∞, ε → 0 and N ( A, ε ) = 4 n .

        n ln ( 4 )   D( A) = lim   n →∞     1  ln    1  n     (2 ⋅ 3 )      n ln 4  = lim   n →∞ ln 2 + n ln 3   =

ln 4 ≈ 1.261859507... ln 3

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 ε n = Cr n for some

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

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   ln( N ( A, ε n ))  DF = lim   ,then A has fractal dimension DF. n →∞ 1  ln  εn  

( )

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

E = {ε n : n = 1, 2, 3,...} be as defined in the statement of the theorem. Let f (ε ) = max{ε n ∈ E : ε n ≤ ε } with ε ≤ r , then f (ε ) ≤ ε ≤ f (ε ) / r and

N ( A, f (ε ) ) ≥ N ( A, ε ) ≥ N ( A, f (ε ) / r ) . Since ln(x) is an increasing positive function of x for x ≥ 1 ,  ln ( N ( A, f (ε ) / r ) )   ln ( N ( A, ε ) )   ln ( N ( A, f (ε ) ) )   ≤ ≤  ln (1/ f (ε ) )    ln(1/ ε )   ln ( r / f (ε ) )  Expression 1 Expression 2 Expression 3

Assume that N ( A, ε ) → ∞ as ε → 0 ; if this is false then the theorem must be true. Expression 3:

 ln ( N ( A, f (ε ) ) )   ln ( N ( A, ε n ) )  = lim    ε →0  ln ( r / f (ε ) )  n →∞  ln ( r / ε n ) 

lim 

 ln ( N ( A, ε n ) )    ln( r ) + ln(1/ ε n ) 

= lim  n →∞

 ln ( N ( A, ε n ) )   n →∞  ln(1/ ε n ) 

= lim 

Expression 1:  ln ( N ( A, f (ε ) / r ) )   ln ( N ( A, ε n −1 ) )  lim   = lim   ε →0 n →∞ ln (1/ f (ε ) )    ln (1/ ε n )   ln ( N ( A, ε n −1 ) )  = lim   n →∞ ln (1/ r ) + ln (1/ ε n −1 )     ln ( N ( A, ε n ) )  = lim   n →∞  ln (1/ ε n ) 

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

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squeeze law of calculus the limit as ε → 0 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 ( s → 0 ). A stat-plot of log ( N s ) / log(1/ 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 A ∈ H (  m ) , 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 (1/2n)  ln( N n ( A))  which intersect the set. If DB = lim   ,then A has fractal dimension DB. n n →∞  ln(2 )  Proof: For m = 1, 2, 3,...,

1 ( N n −1 ) ≤ N ( A,1/ 2n ) ≤ N k ( n ) for all n = 1, 2,3,..., where 2 +1 m

1 k(n) is the smallest integer k satisfying k ≥ n − 1 + log 2 m . The first inequality holds 2

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because the number of boxes of size 1 equal to the number of boxes of size 1 ball has radius of 1

2

n

2 n−1

needed to cover the structure is less than or

2 n−1

needed to cover a ball cover of A where each

. Since each ball of radius 1

2

n

can intersect at most 2m +1 on-

grid boxes of side length 1 2 n −1 , the number of boxes needed to cover the specified ball cover of A is less than or equal to (2m+1)·N(A,1/2n). Figure 15 2

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

2

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

1

15) 1

The second inequality follows from the fact that a box of side s 2

2

2

2

s s s s can fit inside a ball of radius r provided r ≥   +   + ... +   = m   by the 2 2 2 2 2

theorem of Pythagoras. Using s as the highest power of 1/2 that is less than s, let 1 1 1 s ≤ s and s = k ( n ) then k ( n ) ≤ n −1 . Solving for k(n) 2 2 2 m

2 n −1 m ≤ 2 k ( n ) ( n − 1 ) ∗ log 2 2 + log 2 m ≤ k( n )log 2 2 ( n − 1 ) ∗ log 2 m ≤ k( n ) 1 ( n − 1 ) ∗ log 2 m ≤ k( n ) 2 We want the least k that makes this true.

 ln ( N k ( n ) )   ln ( 2 k ( n ) ) ln ( N k ( n ) )  k (n ) The lim  = lim  = D since → 1.   n n k ( n ) n →∞ n →∞ n ln 2 ln 2 ln 2 ( ) ( ) ( )    

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2

1    ln 2 m +1 N n −1   ln N n −1  Since also lim  =D  = lim  n n −1  n →∞ n →∞  ln(2 )   ln(2 )    Theorem 6 with r = 1 2 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.

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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.

Line segment

s1 = 1 2, N ( s1 ) = 2

s2 = 1 4, N ( s2 ) = 4

s0 = 1, N ( s0 ) = 1 N (1) = 1

log ( N ) / log (1 s )

N (1 2 ) = 2 = 1/ (1 2 )

log ( 4 ) / log (1/ (1 4) ) = 1

N (1 4 ) = 4 = 1/ (1 4 )

This relationship holds for every size box so the

and so in general dimension of a line segment is 1, as was

N ( s ) = 1/ ( s ) expected. Unit Square

N(1)=1

s 0 =1, N(s0)=1

N (1 2 ) = 4

N (1 4 ) = 16

log 4 =2 log1/ (1 2 )

log16 =2 log (1/ (1 4 ) )

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

Box Size s 1 1/2 1/4

Number of Boxes N(s) Log 1/ s 1 0.0000 4 0.3010 16 0.6021

s 2 = 1/4, N(s2)=16

Log N(s) 0.0000 0.6021 1.2041

1.4 1.2 1 0.8 0.6 0.4 0.2 0

y = 2x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

The slope of the line is 2 as expected.

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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 boxcounting technique a photo of lightning is analyzed for possible fractal dimension. Figure 16

Figure 17

Begin by making the assumption that the

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

original photograph had a dimension of 1unit

square. 14 out of 16 contain lightning.

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.

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Figure 18

40 out of 64 squares contain lightning and the sides of the squares are now 1 23 = 1 8 the size of the original.

Figure 19

Each square is 1/16 the size of the original and 99 of the squares contain part of the lightning.

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Figure 20

Each square is 1/32 the size of the original and 233 of the squares contain part of the lightning.

Size of Box 0.03125 0.0625 0.125 0.25 0.5

No. of Log (1 / Log(Box Boxes Box Size) Count) 233 1.51 2.37 99 1.20 2.00 40 0.90 1.60 14 0.60 1.15 4 0.30 0.60

By this analysis it appears the lightning has a box-counting dimension of 1.455.

Log-log Graph 3.00

y = 1.455x + 0.2286

2.00 1.00 0.00 0.00

0.50

1.00

1.50

2.00

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 A ⊂  m be bounded. Then the diameter of

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A is diam( A) = sup {d ( x, y ) : x, y ∈ A} . The supremum, least upper bound, might not be attained by any elements x, y ∈ A . Let 0 < ε < ∞ and 0 ≤ p < ∞. Let A denote the set of sequences of subsets

{ Ai ⊂ A}, such that A = ∪i∞=1 Ai . Then define ∞  M ( A, p, ε ) = inf  ∑ ( diam( Ai )) p : { Ai } ∈ A, and diam( Ai ) < ε for i = 1, 2, 3,...  .  i =1  Here we use ( diam( Ai ))0 = 0 when Ai is empty. M ( A, p, ε ) is a number in the range [0, ∞ ] ; its value may be zero, finite, or infinite. Now define M ( A, p ) = sup {M ( A, p, ε ) : ε > 0} Then for each p ∈ [0, ∞ ] there is M ( A, p ) ∈ [0, ∞ ] . [2] Definition 22: Let M be a positive integer and let A be a bounded subset of the metric space (  m , Euclidean ) . For each p ∈ [0, 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 p ∈ [0, ∞ ] , 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 p ∈ [0, ε ) defined above. Then there is a unique real number DH ∈ [ 0, m ] such that

∞ M ( A, p ) =  0

if p < DH and p ∈ [0, ε )   if p > DH and p ∈ [0, ε )  [2]

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

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metric space (  m , Euclidean ) . The corresponding real number DH , occurring in Theorem 8 is called the Hausdorff-Besicovitch dimension of the set A. This number will also be denoted by DH ( 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 DH ( A) denote the

Hausdorff-Besicovitch dimension of A. Then 0 ≤ DH ( A) ≤ D ( A) ≤ m.

[2]

In 1918 Felix Hausdorff introduced the Hausdorff dimension as an extended nonnegative 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 Figure 21

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.

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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.

Name Middle Third Cantor Koch Curve Sierpiński’s Triangle

Topological Dimension

Hausdorff Dimension

Box-counting Dimension

0

0.63

0.62989

1

1.26

1.26186

1

1.585

1.58996 [16]

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 zerodimensional 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 onedimensional, 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.

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

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[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.ohiostate.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

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Appendix

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.

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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. 45 of 57

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.

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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

E C

D

it will have a larger bloom. Α

3 - Iterations

Β

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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 2

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

connection.

two new squares equals the area of the old square. 2 2 2 5) Remember, the Pythagorean Theorem is a + 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).

For a square the number of self-similar squares = (magnification factor)2. 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

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

For a cube the number of self-similar cubes = (magnification factor)3. Two times the side of new cube equals the Letting 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

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Page 2

Letting

m = magnification factor

n = md

n = number of self-similar pieces

log n = log m d

d = dimension of fractal

log n = d log m log n =d log m

then n = md.

For the middle square in figure 1 m = 3 and n = 9. 9 = 3d log 9 = d log 3 log 9 =d log 3 2=d 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 = _______. Using n = m d thus d =

log n log m

solve for d, rounding to the nearest thousandths. d = ________.*

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

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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 3

Figure 2

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.

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Perimeter of Koch Curve

Stage 0

Stage 1

Stage 2

Stage3

Stage

Number

Length of

Total length (in linear units)

of

each

segments

segment

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

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 53 of 57

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 A+  A 9

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 1 + r + r 2 + r 3 + ... =

1 we can see 1− r

that the area under the Koch curve (where stage 1 has an area of A) is:  4  4  2  4 3   1   1  9 A  1 + +   +   + ... = A   = A = A  9 9 9  1− 4 9  5 9 5  

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 Sierpinski triangle.

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. Area of Sierpinski Triangle Area of each triangle removed 0

Total area removed at this stage

Total area removed so far

0

0

0

Number of triangles removed 0

1

1

1/4

1/4

1/4

2

3

1/16

 1 3 2  4 

1  1 + 3 2  4 4 

Stage

Image

3

4 n

You do not need to create these images.

. The area of the Sierpinski triangle is ____________because: ______________________________________________________________ ______________________________________________________________

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. Perimeter Sierpinski Triangle . When discussing the perimeter of Sierpinski 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

Length of each

Total length added

Total length added

sides added added side

at this stage

so far

0

0

0

0

3S

1

3

S 2

S 3  2

3S +

3S 2

2 3 4 n

What is the pattern? What is happening to the perimeter of Sierpinski triangle? ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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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 Sierpinski triangle work was designed for the student to work independently. Answers: 4 Perimeter of Koch curve    3

n

17 stages gives a perimeter > 100

19 stages gives a perimeter > 200 Area Sierpinski 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 Sierpinski Triangle is infinite ∞

 3 n  3S + S     2   n =0

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