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Home , Fractal dimension, Gaston Julia, Julia set, Mandelbrot set, Misiurewicz point

Self‐Squared Dragons: The Mandelbrot and Julia Sets

Timothy Song CS702 Spring 2010 Advisor: Bob Martin TABLE OF CONTENTS

Section 0 Abstract……………………………………………………………………………………………………………….2

Section I Introduction…………………………………………………………………………………………………….…3

Section II Benoît Mandelbrot…………………………………………………………………..…………………………4

Section III Definition of a ………………………………………………………………..……………………….6

Section IV Elementary ………………………………………………………………..………………………….8 a) Von Koch………………………………………………………………..…………………………..8 b) Cantor Dusts………………………………………………………………..……………………..9 1. Disconnected Dusts……………………………………………………………….11 2. Self‐similarity in the Dusts……………………………………………………..12 3. Invariance amongst the Dusts………………………………………………..12 Section V Julia Sets………………………………………………………………..…………………………………………13 a) Invariance of Julia sets………………………………………………………………..…….17 b) Self‐Similarity of Julia sets…………………………………………………………………18 c) Connectedness of Julia Sets……………………………………………………………….19 Section VI The Mandelbrot ………………………………………………………………..………………………..22 a) Connectedness of the ………………………………………………24 b) Self‐Similarity within the Mandelbrot Set………………………………………....25 Section VII Julia vs. Mandelbrot………………………………………………………………..……………………….28 a) Asymptotic Self‐similarity between the sets……………………………………..30 b) Peitgen’s Observation……………………………………………………………..…….…33 Section VIII Conclusion………………………………………………………………..………………………………………34

Section IX Acknowledgements……………………………………………………..……………………………………35

Section X Bibliography………………………………………………………………..……………………………………36

1 Section I: Abstract

Fractals are so complex that they belong in their own independent category of ; fractal . From the time the word “fractal” was created by

Benoît Mandelbrot, this subject has grown from obscurity and unappreciated to parts of our everyday lives. Two of the most famous fractals are created from the same equation. Both the Mandelbrot and Julia sets are formations spawned from

the complex quadratic mapping z1 → z0 + c. Although they both use the same formula, the sets are infinitely different, but still remain closely related.

€

2 Section II: Introduction

Before the discussion about fractals begins, it is most necessary to reveal the man behind the madness. His work is the inspiration for this paper, not only because his work created the basis for study, but because his philosophy on these beautiful beasts motivated me to dive deeper into this field, which applications now span across physics, , medicine, and digital photography to name a few.

Next fractals will be defined carefully to create understanding to any reader.

There are numerous aspects to what defines a fractal, but what is important to note is that every fractal does not have to fit every aspect, but simply it has to match a few of them. This idea becomes clearer in the review of several basic fractals that exemplify these aspects.

After the background on fractals has been covered, the main focus of the paper can begin in its exploration of the famous Julia sets. This survey is mathematical for the sake of accuracy, but also pictoral for the sake of intuition. The pictures are necessary tools to understanding the depth of knowledge that resides within these sets.

In a discussion of the Julia sets it is impossible to avoid discussing the

Mandelbrot set, named after the godfather of fractals. Julia sets are very intricately related to the Mandelbrot set, and by investigating both of them, these similarities should become apparent. Through a rigorous comparison, hopefully the knowledge of both sets will come naturally.

3 Section II: Mandelbrot

Benoît B. Mandelbrot coined the term “fractals” as he was studying the type of geometry that isn’t perfect circles and squares, right angles and forty‐five degree angles, but rather, a geometry that “describes many of the irregular and fragmented around us.”1 This makes sense, because he derived ‘fractal’ from the Latin adjective fractus, meaning “to break”. What began as a journey into an unknown, disregarded field of mathematics quickly turned into the development of a totally new and exciting field of mathematics:

Fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played]… an historical role… in the development of pure mathematics. A great revolution of ideas separates the classical mathematics of the 19th century from the modern mathematics of the 20th… These new structures were regarded… as ‘pathological’…as a ‘gallery of monsters,’ kin to the cubist painting and atonal music that were upsetting established standards of taste in the arts at about the same time. 2

One issue that arose while determining the exact focus of this essay was the fact that

I feared that fractals and their relation to the real world and solid Euclidean

Geometric objects would be too inexact for the purposes of my thesis. However, a quote taken from Mandelbrot’s book explains why this is not an issue:

all pulchritude is relative… We ought not… to believe that the banks of the ocean are really deformed, because they have not the form of a regular bulwark; nor that the mountains are out of shape, because they are not exact pyramids or cones; nor that the stars are unskillfully placed, because they are not all situated at uniform distance.3

This attitude actually excited me and encouraged me more in investigating this field.

As a student of Science that was never too fond of the mathematics side, this realization came as a welcome relief. It also offered a fresh perspective compared to the one that I’ve usually experienced in my short mathematical

4 experiences. However, even though this study does not involve 100% and formulas, there are many definitions that must be clarified before the study begins.

5 Section III: Definition of a Fractal

Mandelbrot outlines three important concepts in his book, before he divulges into the deeper intricacies of fractals. First, the definition of a fractal, from the man who coined the term: “is by definition a set for which the Hausdorff‐Besicovitch

4 strictly exceeds the topological dimension.” Topological dimension, DT

(aka Lebesgue Dimension) is the minimum number of it takes to describe a shape. So, for a , the topological dimension would be 0, for a line: 1, for a plane:

2, for a cube: 3 (height, length, width). (D), also defined informally, for self‐similar (meaning each instance of the object has identical copies of itself at different scales) objects would be: D = log(number of pieces you split each )/log(each level of magnification per iteration).5 Lauwerier gives a more precise definition:

We select an arbitrarily small measurement unit a, the yardstick. Next we measure the length of the meandering line by approximating it as closely as possible with a bent line made up of equal line‐segments of length a. If we suppose the yardstick is used N times, so that the total length measured is N a, then according to Mandelbrot’s definition the “” is given by6

logN D = lim a →∞ 1 log a

Hausdorff dimension has many other names, including fractal dimension and similarity dimension. Put in a mathematical inequality: D > DT. € Although Mandelbrot originally gave the term “fractal” a very specific definition, through the years the exact definition of fractal has become a little more lax. Many associate the term fractal with self‐similarity. An object is self‐similar if it

6 looks approximately the same on small scales as it does in large scales. However self‐similarity is not limited to a strict definition of simply looking the same on different scales. Self‐similarity also can fall across translation, meaning that you don’t see exact copies, but smaller, rotated, copies as well. More discussion about self‐similarity will follow in relevant sections.

Fractals are also known to have simple, recursive definitions. This adds to their because such simple definitions end up creating very intricate structures. They also have the distinguishing feature of being indescribable through simple Euclidean Geometry (e.g. squares, circles, etc.) Sometimes these structures can be infinitely complex, such as the Mandelbrot set, discussed in Section VI. This infinite complexity can border on chaos.

7 Section IVa: Von Koch Snowflakes

The von is a common elementary fractal that is used to familiarize people with the world of fractals. Its simple iterative rules make it easy to conceptualize and its properties are very distinctive of fractals. If you look at

Figure 1 below:

FIGURE 1 7 you will see the von Koch replacement rule in 4 , starting on a straight line.

Each iteration takes the middle third of a straight‐line segment, and then adds two more pieces of equal length which are placed at an angle, creating two sides of a triangle. Every time you follow this process, the length of your line is increased by

4/3. So, if each iteration increases the line by this amount, infinitely many iterations will create a line of infinite length, which Mandelbrot names a von Koch arc. This should also be noted that this line is also bounded in a finite .

8 According to our definition of dimension, DT = log(4)/log(3) ≈ 1.26, and D =

1, meaning DT < D. So, by definition, von‐Koch arcs are fractals. “It is continuous, and it has no definite tangent anywhere – like the graph of a continuous without a derivative.”8 This feature makes the von Koch arc a very distasteful curve to conventional mathematicians, who are recorded as “turning away in fear and horror from this lamentable plague of functions with no derivatives.”9

If the von Koch replacement rule is iterated upon a triangle, your results become the well‐known von Koch snowflake. Just like the von Koch arc, when the iteration is run infinitely many times, the border length becomes infinite, which for this snowflake also means that the becomes the circle that you draw around the first triangle. Basically, it approaches that bound.

FIGURE 210

One of the early criticisms of the study of Fractal Geometry that Mandelbrot started was that fractals were “pretty pictures” and nothing more. However, taking the von‐

Koch fractal as an example, fractals have become surprisingly useful in a variety of practical purposes.

Section IVb: CANTOR DUSTS

While Von‐Koch snowflakes are surprising in its infinite length and visual appeal makes it interesting, there are other fractals that disappear with further

9 iterations. The , or Cantor Dusts, is an example of such a set. Georg

Cantor had the idea for a that initiates from [0,1]. “For each number in this interval, there is a corresponding point in the Cantor Set… Thus, the of the Cantor Set must be at least as large as the cardinality of the interval.”11

The set is closed because it includes the endpoints. To create the set, simply divide the set into thirds, and remove the middle third. At the first step of this iteration, you will remove the ]1/3,2/3[, open brackets signifying the exclusion of the endpoints. If this iteration is continued infinite times, the result, according to Mandelbrot, is the Cantor fractal dusts. Figure 3 below illustrates the first 3 iterations.

FIGURE 312

The Cantor set is a very good example of a fractal because it’s Hausdorff dimension is D = log(2)/log(3) ≈ 0.63, while it’s topological dimension DT would be

0. Since DT < D, and the set is a fractal. It is also self‐similar (each iteration is a third the length and translated) which makes it a good example of a fractal. Cantor dusts also have another interesting feature that make them worthy of note. Cantor

10 sets, when iterated towards , become totally disconnected. As seen in Figure

3, the line that starts as completely connected quickly becomes disconnected.

Section IVb 1: Disconnected Dusts

In topology, a set is called connected if it cannot be decomposed into two

disjoint, non‐empty subsets (which are both open and closed in the topology of the

set).13 Another type of connectedness is called pathwise connected. This means that

a continuous path that is entirely within the set can connect any two points of a

set.14 On the other hand, the inverse of a connected set is a totally disconnected set,

which means that a set is connected components (i.e. maximal connected subsets)

are single points.

When observing the higher iterations of the Cantor Set, nicknamed The

Cantor Dusts, it is clear that the line segments become smaller and smaller. What

cannot be seen by the naked eye is that the line segments eventually become points.

At this level of iteration, the Cantor Dusts become totally disconnected. The proof

follows from Nelson:

Proof: Fix any ε > 0 and point p ∈ C. Let n ∈ N be sufficiently large such that 1 < ε. Then, p is guaranteed to be in one of the intervals (In for some n ∈ N ) that 3n 1 make up C, each of length € € . The endpoints of the Cantor set in this interval€ are 3n infinite in number, and all contained in the open interval (€p −ε, p + ε). So, p is a € cluster point of C,Mε (p) containing an infinite number of points. And since we are considering any p ∈ C, C is perfect. Furthermore, this interval In is closed in and € c so in the Cantor set C as well. Since In = C \ In consists of a countable number of closed intervals, it is itself closed. We can then represent C as the disjoint union€ of € c two clopen sets, ( C ∩ In ) and ( C ∩ In ), the result being that the Cantor set C is totally €disconnected. 15 € 11 € €

This property is important to our study of the Julia sets, which will be continued in

Section V.

Section IVb 2: Self‐Similarity in the Dusts

The idea of self‐similarity is an intuitively simple one to grasp when you view the Cantor Dusts. Take a look at Figure 3. If you view the set from the first line segment and down, you can see the clearly forming: each next iteration has it’s middle third removed to create the following iteration. To show how the set is self‐similar, simply remove all the pieces to the right from the picture and you will have a smaller, similar, set of Cantor Dusts. If you elongate every piece of the remaining side by 3, you will have your original Cantor Dusts. This example shows us quite easily that the set is self‐similar through a linear transformation, which is not the case for all fractals.

Section IVb 3: Invariance amongst the Dusts

This characteristic of self‐similarity also helps visualize another characteristic of the Cantor Dusts, which is invariance. A set will be if it does not change under certain transformations. As a simple example, we claim that the Cantor Dusts are invariant. If you take any point within the Cantor set and multiply it by 3, you will find another point that lies within the Cantor set. This

12 property, as well as the previously discussed properties of the Cantor set, is crucial to understanding the Julia sets, which share a lot of these properties.

13 Section V: Julia Sets

Gaston Julia was a mathematician who studied mathematics during the late

19th and early 20th century. At the humble age of 25, his most famous paper,

"Mémoire sur l'itération des fonctions rationnelles" was published in the famous

French mathematics journal, Journal de Mathématiques Pures et Appliquées. This paper described a method of iteration of a . Although his ideas were published in 1918 and were highly praised at the time, it wasn’t until Benoit

Mandelbrot discovered the paper and began using Julia’s findings in his own studies that Julia’s work became well known.

Julia sets are a special type of fractal that is so intricate and complex looking that Mandelbrot originally deemed them “self‐squared dragons”. A self‐squared dragon is another name for quadratic fractals. Perhaps Mandelbrot used this name to poke fun at the critics that were calling his fractal discoveries “monsters”, but the name is actually quite suitable once you begin to see certain computer‐generated variations.

FIGURE 416

These sets are called self‐squared because the iterative rule used to create them is simply a quadratic mapping: 14 zn+1  zn2 + c FIGURE 5

Where z and c are complex numbers. This simple mapping, iteratively repeated results in a of complex numbers:

2 2 2 2 2 2 z0  z + c  (z + c) +c ((z + c) + c) +c…

Julia studied many different types of rational expressions, but this survey focuses on the most famous mapping; the one shown above which closely relates to one of the most famous fractals in history, the Mandelbrot Set, discussed in detail in Section VI. All above variables are complex numbers in the form a + bi, where a and b are real numbers. The c is a constant that is the key to the beauty and complexity of the . It does not change during iteration, but each separate Julia set has a separate c value.

Although c holds the key to the Julia sets amazing visual structure, the starting value of z is the variable that defines whether a given point will be a part of the Julia set. As for the question, how do the values of c and z affect the set, I find it most useful to use pictures to help aid intuition.

FIGURE 3a17 FIGURE 3b

15

FIGURE 3c FIGURE 3d

FIGURE 3e

All these Julia sets are created using unique c values. The constants used to generate each figure are located in the bottom field. An important aspect to these separate Julia sets is that there exist infinitely many versions of the Julia set.

Because c is a complex number, and there are infinitely many complex numbers, there are infinitely many Julia sets.

To create the visual “dragons” such as in Figure 4 seen above, the Julia Set must be iterated upon a . During the infinite iterations of the quadratic mapping seen in Figure 5, the chosen constant of c will cause the Julia set to display one of two types of behavior. Heinz‐Otto classifies this dichotomy of the

Julia Set:

16 1. Either the sequence becomes unbounded: the elements of the sequence leave

any circle around the origin

2. Or the sequence remains bounded: there is a circle around the origin which is

never left by the sequence18

Any value that falls within the first category is classified as the escape set (E). These values trend toward infinity. The second category is classified as the prisoner set

(P), or the set of values that do not tend toward infinity, and stay within a limits.

Neither set can be empty. The escape set E can be precisely defined: Ec = {z0 : |zn| 

∞ as n  ∞} while the prisoner set P can be precisely defined as Pc = {z0 | z0 ∉ Ec}.

The Julia set is defined as the between the prisoner set and the escape set.

Perhaps it would be easier to conceptualize the Julia set through a visual representation. ’s ideas were far ahead of his time, and he didn’t live long enough to see and computer graphics advance to the point where they could iterate upon these sets. Today, there are endless applications and web‐ applets that draw the set for you and let you zoom and explore the set yourself.

Clearly, humans doing calculations by hand would never be able to calculate infinite iterations on all possible points. Even a computer cannot calculate infinite numbers and infinite points. So, programs creating images of the Julia set are set to compute a certain number of iterations before declaring a point as part of the escape set or the prisoner set. Once the set is declared, then the point is designated a color. The most elementary scheme would be black and white; black being part of the prisoner

17 set and white being part of the escape set. You can see the clear distinction between the sets in Figure 6 below.

FIGURE 619

When looking upon these “dragons” is it clear that they are in fact, fractals? If you look closely at Figure 6, it is clear that there are many smaller copies of the twisting aspect of this Julia set, as well as smaller copies of the whole in the spiked corners of the Julia set

Section Va: Invariance of Julia Sets

Determining the invariance of the Julia sets is a multiple‐step process. First, we must discuss the Julia set’s iteration. The Julia set mapping is created from the

2 simple iteration on z0  z1 + c, meaning each complex number z is calculated as the square of a previous complex number, plus a complex constant c. So, we could re‐ write this formula as our new complex number w = z2 +c. We can isolate the variables to one side and get: z2 – w + c = 0. Because we know that z is a complex number, we can isolate it to one side and then take the complex to get:

18 z1,2 = ± w − c . Because this solution could be positive or negative, that means there

are two solutions, or preimages (z1, z2) for each w, with the exception of w = c. This

€ gives us two transformations:

w → + w − c w → − w − c FIGURE 7

€ 2 These transformations are derived from our original Julia set mapping, z → z + c , € which signifies that any point in the Julia set w undergoing these transformations

will create another point on the Julia set. In other words, the Julia set is an € invariant

set with respect to the inverse transformations in Figure 7. This also means that the

Julia set is an invariant set with respect to z → z2 + c .20 Our original mapping is

classified as forward iteration because you start at the beginning and keep iterating

until your reach the end (or in this case, a maximum number of iterations). € The

mappings we created in Figure 7 are method of backwards iteration, and because

the Julia set is invariant under both these conditions, the Julia sets are said to have

complete invariance.

Section Vb: Self‐Similarity of Julia Sets

Because Julia sets have complete invariance implies that the “global structure

of the Julia sets must also appear in the images and preimages of the Julia set, which

explains the apparent self‐similarity.”21 Unlike the Cantor Dusts similarity

described in Section IV‐b2, this self‐similarity occurs through a non­linear

transformation. In simpler terms, self‐similar co pies of the set occur in distorted,

twisted versions of the whole. The Best visual representation of this idea is Douady

19 and Hubbard’s rabbit, seen in Figure 8 below. Each boxed section is a replica of the others, and the line in the middle separate two equal halves that are translated and flipped.

FIGURE 822

This image has been kept large so you can see the similarities in its structure.

Section Vc: Connectedness of Julia Sets

So far, we have discussed some interesting aspects of the Julia sets. All these points culminate into the most important characteristic of the Julia sets: their connectivity. A definition of connectivity is offered in Section

The founders of the original theories that lead to the discovery of the Julia set, Fatou and Julia, came up with the structural dichotomy of the Julia sets. This stated that:

20 1. The Julia set is a Cantor set if and only if, the iteration of the critical point 0

leads to infinity (in )

2. The Julia set is one piece (i.e. connected) if and only if the iteration of the

critical point 0 is bounded.23

The critical point is the x‐coordinate on your graph, and the y‐coordinate is the critical value. The fate of the critical point, or the critical determines if your

Julia set belongs to either one of these two categories. Whether a Julia set becomes a totally disconnected Cantor set or a connected set is important in its relation to the

Mandelbrot set., which will be discussed in Section VII.

FIGURE 10a FIGURE 10b

As you can see in Figures 10a and 10b, Julia sets can display very different types of behavior. In 10a, we have a Julia set that is totally disconnected. At first glance, the structure might seem to have some islands scattered about, but if you zoom into the drawing close enough, and with enough computational accuracy, you will find that the set is made up of distinct, disconnected points and nothing else. It

21 carries the same qualities of the Cantor dusts discussed in the earlier section. On

the other hand, Figure 10b displays the opposite type of behavior in that it remains

connected. Simply put, the entire set is one piece, and all points within the set can

reach each other while travelling through the set.

The following from Julia and Fatou: “Let Ωp denote the set of critical

points for a polynomial P. Then: € • Ωp ⊂ K p ⇔ J p is connected 24 • Ωp ∩ K p = ∅ ⇒ J p is a Cantor set

This theorem states that every Julia set must be either connected, or totally € € disconnected (i.e. a Cantor set). The proof of the theorem involves the closed curves

surrounding filled‐in, connected Julia sets. For this proof, not contained here for its

mathematical complexity, look it up in Devaney 1989 (Reference 24). This proof is

important not only to the connectedness of the filled‐in Julia sets, but the same

strategy is used to prove the connectedness of the Mandelbrot set.

22 Section VI: The Mandelbrot Set

When discussing the Julia set, is it inevitable to gravitate towards discussion

of the Mandelbrot Set. If the Julia set was the parent, the Mandelbrot set is most

definitely the child. Now that we have extensively explored the Julia sets, and

covered many different interesting features of said set, what is the next step? From

the last section, we have resolved that there are infinitely many Julia sets, and that

all Julia sets fall into two categories:

1. Connected

2. Totally disconnected (i.e. a dust)

This is called the structural dichotomy of the Julia sets, as labeled by its discoveries,

Fatou and Julia. However infinitely interesting this fact may be, its usefulness and

practicality was in question until decided to explore this

dichotomy is 1979. In both of these related sets, the same iteration is used:

z → z2 + c . Recall that for the Julia sets, we keep c fixed and iterate upon the

complex z to determine whether the critical point trends towards infinity. In the € case of the Mandelbrot set, we keep the initial z0 fixed, and each critical point is

changed according to c. Mandelbrot used his understanding of the Julia sets and

computer graphics to generate a pictoral representation of the dichotomy in the

complex plane. Through his experiment, the Mandelbrot set M was born.

M = {c ∈ C | Jc : is_connected} 2 M = {c ∈ C | c → c + c → ...remains_bounded}25

€ € 23 Mandelbrot graphed his idea on the complex plane from ‐2 to 2, because when |c| > 2, the critical point escapes to infinity. In other words, the complex iteration z will be attracted to infinity, and be part of the escape set E. This proof can be found in [Peitgen 794]. This also means that it falls under the second of the two categories listed above, and the resulting Julia set will be a Cantor dust, a totally disconnected set. The Julia set contains a bounded critical point for the interval [‐2,

0.25], meaning every complex number within this range creates a connected Julia set. These connected Julia sets are what make up the Mandelbrot set.

FIGURE 1126

To really appreciate the complexity of the Mandelbrot set, you have to see it.

Creating a visual representation of the set entails counting the number of iterations that it takes for the mapping to escape to infinity for each pixel on the screen. By calculating the bound for each pixel, you are checking to see if for that c value, the quadratic iteration escapes to infinity or not. Instead of calculating infinite iterations (which would be quite impossible, even on a computer), programmers usually use delineate some sort of number of maximum iterations where the point will be assumed to trend towards infinity and assign it a color. In the case in Figure

24 10, those numbers that follow this trend (i.e. are in the escape set E) are colored dark blue and those that are bounded, (i.e. in the prisoner set P) are colored black,

The additional colors are included to make the smaller intricacies of the Mandelbrot set more visible and aesthetically pleasing.

At the most basic level, the Mandelbrot set is everything that falls in the black area of figure 11.

Section VIa: Connectedness of the Mandelbrot Set

The intricate pictures of the Mandelbrot Set are as beautiful as they are complicated. It is this infinite complexity that makes the set so intriguing. One important aspect of M is that it is connected. Originally, after Mandelbrot printed out the first image of the Mandelbrot set, he initially thought that M was disconnected, because outside of the main “bulbs”, there seemed to be some scattered dusts. The error was not the mathematician, but the machine. Printers just could not print at resolutions high enough for the connectedness to be visible.

This is an understandable mistake once you see what Mandelbrot saw so many years ago:

FIGURE 12

25 However, it is in fact simply connected, as to the definition offered in Section

IV‐b2. Douady and Hubbard proved this long ago, in their famous paper “Iteration des polynomes quadratiques” or “The Iteration of Quadratic ”. The proof states that “the encirclement of the Mandelbrot set always generates domains which are bounded by circle‐like curves. If the encirclement is properly manufactured it can be shown that the bounding curves are in fact equipotentials of

27 the Mandelbrot set.” More precisely: “Let VM (r) denote the closed, simply connected domain bounded by (r). Then: M V (r) . Hence, M is connected.”28 ΓM =  M r>1 € Section VIIb: Self‐Similarity within the Mandelbrot set € € This aspect of the Mandelbrot set is truly interesting when you observe the smaller islands that are surrounding the Mandelbrot set. The next set of pictures was obtained using the Macintosh Power Fractal program.

FIGURE 13a

26

FIGURE 13b

FIGURE 13c

FIGURE 13d

27 Figure 13a is clearly the full Mandelbrot set. The rest of the pictures are taken from different surrounding the main “bulb”. Douady and Hubbard have a theorem that there are infinitely many copies of the Mandelbrot set within the Mandelbrot set, here we see a few of them. The important thing to note is that the Mandelbrot set is quasi‐self similar: it is does not fit a simple definition of self‐similarity. Rather, the Mandelbrot set is described as being asymptotically self‐similar at Misiurewicz points. These points have a special relation to the Julia set as well, and will be discussed in Section VII‐b.

28 Section VII: Julia vs. Mandelbrot

Now that both the Julia sets and the Mandelbrot set have been defined and

analyzed to the point of familiarity, let us examine the differences, similarities, and

relationships between two of the most famous fractals in existence. The first

observation, that has already been mentioned, is that the two sets both use the same

iteration ( z " z2 + c ). How then, are the structures of the sets so different?

The answer is as simple as the iteration itself. Recall that Julia sets are

measured iterations of the above equation for constant ! c values. For every unique c

value, there will be a unique Julia set. This iteration, with a constant c, is mapped on

the complex plane. Your complex plane obviously holds complex values, and for

each of these values is a different starting z0 value. You iterate upon the different z

values with the same c. So you will get iterations that look like this:

z → z2 + c → (z2 + c)2 + c...

The Mandelbrot set takes the opposite approach. The iteration uses a € standard starting z0 =0 value for every iteration. It will iterate, instead, on a

variation of c. So, on the complex plane, again imagine all of the complex numbers

that fall on it. Each of these points will be used as a separate c value for the

iteration, and each point will either iterate towards infinity or remain bounded.

Since we start at 0, iterations will look like this: 0 → 02 + c → (02 + c)2 + c...

This simple variation causes an aesthetically pleasing result. The Mandelbrot

set is quite beautiful, and its beauty is accented by € its intricacies. Firstly, the entire

Mandelbrot set is “a road map for Julia Sets.”29 Defined casually, but accurately,

29 every point on the Mandelbrot set mapping corresponds to it’s own unique Julia set.

Intuitively, every Mandelbrot point is created using a different c value. If every Julia set is created using a different c value, then the intuition should follow that

Mandelbrot points create Julia sets. To aid intuition, the following figure shows the correspondence between the sets.

FIGURE 1430

Each Julia set seen surrounding the Mandelbrot set corresponds to the c point indicated. If there exists more interest in this relation, there are many, many web applets that allow users to click on a Mandelbrot set and generate the corresponding Julia set.

These Julia sets also follow a trend with regards to the Mandelbrot set. For every Julia Set created with a c value that falls within the Mandelbrot set, you have a connected Julia set. If the c value used to create a Julia set falls outside the boundaries of the Mandelbrot set, you then have a totally disconnected Julia set.

Observe Figure 14. On the left, you can see a c point that is chosen outside of the

30 Mandelbrot’s barriers. This set is clearly a dust, with its barely visible portions and very gray appearance.

Section VIIa: Asymptotic Self‐Similarity between the Sets

The relation between the two types of structures does not stop at their iteration. As shown earlier, the Mandelbrot set has an infinite level of detail, only limited by the calculations a computer can complete in a timely manner to display images of great detail. This property is shown when examining the smaller

Mandelbrot sets that lie within the details of the full set. At the minute level, magnification levels 300,000,000 fold from the original, we can find points that are indistinguishable from magnifications of related Julia sets. Again the easiest way to understand these concepts is through pictures:

FIGURE 15.1‐a FIGURE 15.2‐a

31

FIGURE 15.1‐b FIGURE 15.2‐b

FIGURE 15.1‐c FIGURE 15.2‐c

FIGURE 15.1‐d FIGURE 15.2‐d

Clearly the first two figures (15.1‐a and 15.2‐a) are similar. They both show the same spiral pattern and have the same smaller buds surrounding them. The sequence in the left column is from the seahorse valley of the Mandelbrot set. The

32 right‐column sequence is the Julia set created from the c value (‐0.7746899 +

0.1242248i) at the center of Figure 15.1‐a. Each time you move down an image, the frame has zoomed out slightly. All figures were drawn with Mac OSX software

Fractal Domains 2.0.9 (© Dennis C. De Mars).

This similarity is not self‐similarity that exists in more simple fractals structures like the Cantor Dusts. Their similarities arise through more complex mathematics than simple rotation and magnification. For this specific type of similarity, we assign the term asymptotically self­similar. “We call I asymptotically self‐similar at the point z0 if there are:

• A complex scaling factor p, called multiplier, with |p| > 1

• A small radius r >0

• And a limit object L (a subset of the complex plane) which is self‐similar at

the origin

n 31 Such that the relation lim Dr (0)∩ p (I − z0 ) = L ∩ Dr (0) holds” , where p is a n →∞ complex number called the scaling factor, and Dr(z) is a centered at z. The more extensive proofs can be found € in Tan Lei’s paper on Misiurewicz points32.

Misiurewicz points are very complex mathematically, but visually it is quite simple to see the correlation between the sets at Misiurewicz points if you observe the pictures. The above sequence, is in fact, one of these points. Interestingly enough, these points exist all over the boundary of the Mandelbrot set; they are

“dense at the boundary of the Mandelbrot set. This means that if we take any point

33 on the boundary of M and an arbitrarily small disk around that point, then there exists a in that disk”33. A summary of Tan Lei’s proof says that

• The Julia set Jc and the Mandelbrot set are both asymptotically self‐similar in

the point z = c using the same multiplier p.

• The associated limit objects Lj and LM are essentially the same; they differ

only by some scaling and a rotation (LM = λLj, where λ is a suitable complex

number)

Section VIIb: Peitgen’s Observation

Peitgen also finds a similar similarity between the sets. For certain values of c, and certain magnifications, M and Jc produce very similar images. Although the images are not as identical as Misiurewicz points, there is most definitely similarity between the two sets. These figures were also drawn using Fractal Domains:

FIGURE 16a FIGURE 16b

34

FIGURE 16c FIGURE 16d

The similarity is blatant, here seen between the Mandelbrot set on the left and Julia set created (16b) using the c = ‐0.745429 + 0.113008i from the center of Figure 16a.

The arrows display where the zoomed picture originates from, the same point on the complex plane in both figures, which is the same value as the c given earlier.

35 Section VIII: Conclusion

Through this survey, we have discussed the man who I’ve deemed “The

Godfather of fractals”, Benoît Mandelbrot. His studies have opened up the entire field of geometry to include these new structures. Hopefully this paper was effective in helping readers experience their beauty mathematically and visually.

From the most simple fractals like the Cantor Dusts, to more advanced fractals like the Mandelbrot set, all these beautiful figures have most definitely changed my opinion of mathematics.

Through the examination of the Julia sets and the Mandelbrot set, we have just begun to scratch the surface of the intricacies of fractals. Up to this point, fractals have been used in car antennas, wireless cell phones, as well as US Marine camouflage design. The practicality of fractals, like their structure, is infinite.

36 Section IX: Acknowledgements

I’d like to thank my family for supporting me in every way throughout my life. If it wasn’t for them, I would most definitely not be able to get this far in life, let alone college. Second, I’d like to thank Bob Martin, my thesis advisor, for having patience with me as we struggled to find a topic. Eventually we found something that peaked our interests, and I appreciate his patience and guidance throughout this process. Finally, I want to thank the Middlebury Computer Science department for allowing me to study the wonderful world of computers. Thanks and I hope to keep in touch with you all.

37 Section X: Bibliography

1 (p1) Mandelbrot, Benoit B. “The Fractal Geometry of Nature” W.H. Freeman and Company, 1983 2 (p3) Mandelbrot 3 (p6) Mandelbrot 4 (p15) Mandelbrot 5 (p19)McGuire, Michael. “An Eye for Fractals” Addison‐Wesley Publishing Company Inc. RedWood City, CA 1991 6 (p33) Lauwerier, Hans “Fractals” Princeton University Press 1991 7 Falconer, K. 1990, Fractal Geometry: Mathematical Foundations and Applications (Chichester: John Wiley and Sons). 8 (p35) Mandelbrot 9 (p36) Mandelbrot 10 Jeffery Johnson, COMPUTER WEEKLY, March 30,1989 via: http://www.fortunecity.com/emachines/e11/86/newmath.html 11 (p75) Peitgen, Heinz‐Otto “Chaos and Fractals: New Frontiers of Science” Springer 2004 12 Noel, Griffin “Cantor Dust” Taken 04‐28‐10 via: http://spanky.triumf.ca/www/fractal‐info/cantor.htm 13 (p803) Peitgen 14 (p803) Peitgen 15 Nelson, Dylan “The Cantor Set – A brief Introduction” UC Berkley 16 Eequor, Wikipedia, November 25, 2004 via: http://en.wikipedia.org/wiki/Julia_set 17 Julia Set Applet via: http://www.shodor.org/interactivate/activities/JuliaSets/ 18 Peitgen 19 Ashlock, Daniel. University of Guelph. April 2010 via: http://eldar.mathstat.uoguelph.ca/dashlock/ftax/Julia.html 20 (p823) Peitgen 21 (p824) Peitgen 22 Winter, Dale, “Patterns in Iteration and the Graph of a Function” http://www.math.lsa.umich.edu/mmss/coursesONLINE/chaos/chaos2/index.html 23 (p834) Peitgen 24 (p80) Devaney, Robert L “Chaos and Fractals: The Mathematics behind the Computer Graphics” American Mathematical Society 1989 25 (p843) Peitgen 26 Beyer, Wolfgang “Mandelbrot Set” 2005 via: http://en.wikipedia.org/wiki/File:Mandel_zoom_00_mandelbrot_set.jpg 27 (p849) Peitgen 28 (p91) Devaney 1989 29 (p855) Peitgen

38

30 Miqel, taken 04‐28‐10 via: http://www.miqel.com/images_1/fractal_math_patterns/mandelbrot‐set/ 31 (p885) Peitgen 32 Lei, Tan “Similarity between the Mandelbrot set and the Julia sets” Springer‐ Verlag 1990 33 (p886) Peitgen

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