fielder - fractal geometry

8/13/2019 Fielder - Fractal Geometry

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Fractal Geometry

James Fielder

Why is geometry often described as "cold" and "dry"? One reason lies in its inability todescribe the shape of a cloud, a mountain, or a tree. Clouds are not spheres, mountains notcones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straightline.

The opening of "The Fractal Geometry of Nature" by Benoit Mandelbrot

Take a point called Z in the complex planeLet Z1 be Z squared plus CAnd Z2 is Z1 squared plus CAnd Z3 is Z2 squared plus C and so onIf the series of Zs should always stayClose to Z and never trend awayThat point is in the Mandelbrot Set

Mandelbrot Set by Jonathan Coulton

Contents

1 Introduction 2

2 Defining Fractal Geometry 22.1 Structure on arbitrarily small scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 The four types of Fractal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.4.1 Iterated Function Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4.2 Recursively Defined Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4.3 Random Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4.4 Chaotic Attractors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 The History of Fractal Geometry 53.1 Set Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.1 The Cantor Ternary Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 The Weierstrass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 The Koch Snowflake. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 The Sierpinski Triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 The Julia and Fatou set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 Benoit Mandelbrot, his set, and the essay that started it all . . . . . . . . . . . . . . . . . . . 93.7 After Mandelbrot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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4 Generating images of Fractals 10

5 Fractals in Art 11

6 Fractals in Science and Nature 126.1 Fractals in Weather analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Fractals in Theoretical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 Fractals in Biological systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.4 A few other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Conclusions 13

1 Introduction

Why are fractals just so fascinating? These enigmatic and beautiful objects have been capturingpeoples imagination for many years, and being mathematically minded myself, it is no surprise that I

fell in love with them when I first laid eyes on them. Fractals are fascinating to look at, and even moremesmerizing to play with on a computer. In this project, I hope to demonstrate how visually andmathematically pleasing fractal geometry is. I hope to show you how the stunning images of fractalsthat are available today, particularly on the Internet, are generated and rendered. Finally I hope toshow why fractals are more than just mere mathematical curiosities; that they have important andpowerful uses in fields as far reaching as theoretical physics to medicine and biochemistry. Also toshow that fractals are still an emerging field with exciting developments to be found.

2 Defining Fractal Geometry

Before any of the discussion of the properties and interesting aspects of fractal geometry, we must

understand exactly what a fractal is, and what these definitions actually mean. There are 3 definingproperties of fractals that all mathematicians seem to be able to agree on. [25]

A fine structure on arbitrarily small scales, which makes it very difficult if not impossible todescribe it in terms of standard Euclidean geometry.[25]

Self-similarity throughout the structure of the fractal. A Hausdorff dimension which is greater than its topological dimension. [17]

In order to fully understand these, we must consider each one of them separately.

2.1 Structure on arbitrarily small scales

The most direct and succinct definition you will commonly hear of a fractal is that one can zoominfinitely on the shape, and still see an ever increasing amount of detail. [16] This captures manypeoples initial fascination with fractals, as it seems strange that the detail you are seeing can go onforever, to scales that would make the size of the universe seem tiny, and still continue.

This leads onto self similarity.

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2.2 Self-similarity

As one continuous to explore a fractal, structures that have been seen before will begin to repeatthemselves. For example, zooming in on certain places in the Mandelbrot set will lead to objectswithin the set that look like the set again. This self-similarity is one of the main factors that givesfractals their unique properties. There are three types of self similarity: Exact self-similarity, quasiself-similarity and statistical self-similarity.

Exact self-similarity: The strongest form of self-similarity, this implies that the fractal looksidentical at all scales. This is usually something that fractals defined by iterated function systemshave. A good example is the Koch snowflake. [25]

Quasi self-similarity: Fractals like this are not exactly the same on all scales, but have featureswhich recur throughout the object. Fractals defined by recursion relations usually exhibit thisbehaviour, and a good example of this would be the Mandelbrot set. [20]

Statistical self-similarity: This is usually exhibited by random fractals, generated by processes

like random walks (brownian motion is a good example of this). For this self-similarity, onlystatistical measures or numerical quantities calculated (for example, fractal dimension) from thefractal are preserved from one scale to another. These generally do not look self-similar. [25,17]

2.3 Hausdorff dimension

The Hausdorff dimension of an object is widely referred to as the fractal dimension of a shape.While there are other notions of dimension, the Hausdorff measure is regarded as being the mostrelevant in fractal geometry.

Take a fractal set in a metric space1 and consider covers of it by balls 2. Measure the volume ofsuch covers by summing the volume of the balls and pretending each ball has dimension d, i.e. that

the volume of a ball of radius r is

d2

( d2

+ 1)rd whereis the gamma function. NoteVdthe minimal

volume of such covers. [7,12, 17]Then, for somed0, the Vd0 is finite, and for alld< d0(resp.d> d0) Vdis infinite (resp. zero). This

valued0is the Hausdorff dimension of the set and therefore the Hausdorff dimension of the fractal.Since the function to calculate the volume of the spheres is continuous, the Hausdorff dimension

can take any positive value. This is unlike the topological dimension of a fractal, which is more likethe concept that one generally has of dimension, which is the number of points required to fully defineany place in that space. [7,12,13, 17,5]

2.4 The four types of Fractal

There are three main types of fractals, which exhibit the three types of self-similarity. These are:Iterated function systems, recursively defined fractals and random fractals. It is also worth consideringchaotic attractors. Let us examine these four separately.

1A space endowed with a distance map, for example any Euclidean space n forn.2The ball of center P and radius r is the set of points of distance to Pless thanr; this concept generalizes that of discs in

the plane.

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2.4.1 Iterated Function Systems

Iterated function systems were primarily introduced by the mathematician John E. Hutchinsonbut were popularised by Michael Barnsley in his book "Fractals Everywhere". The idea of an iteratedfunction system is that one takes a space on which the functions will act, for example 2 being theeuclidean plane, and then some functions f

n

wheren. The next points are calculated by applying

one of the functions to the last points:

(xi+1,yi+1) = f(xi ,yi )

[5]Wherei is one of the iteration steps, and is one of the functions fn which is chosen randomly.

The functions will usually have a probability of being chosen in the "chaos game" associated withthem.

Many fractals which originally were not given as iterated function systems can be translated intothem, for example, in the start of Barnsleys "Superfractals" book he gives an example of an iteratedfunction system which generates the Sierpinski triangle. The most famous example of an iterated

function system is probably Barnsleys fern, which looks exactly as the name would imply, and is astriking example of nature and fractal geometry coinciding. [5]

2.4.2 Recursively Defined Fractals

Recursively defined fractals are the most well known form of fractals, and are the kind that in-cludes the Mandelbrot and Julia sets. Images of such fractals are most commonly in the complexplane, , as here processes like exponentiation can produce interesting images and will not alwaystrend away to infitity for obvious values. Recursively defined fractals are of the form

f :XX

whereX is the space and f is the function that is being applied. In terms of points, these are usu-ally polynomial equations in the complex plane that are repeated, with the addition of a constant.Symbolically this is:

f :zz k + cwhere fis the function,z is a complex number,k is some power 3 to raise the number to andk.

These are most commonly plotted using escape time algorithms, which we will discuss later. [25,16, 17]

2.4.3 Random Fractals

Random fractals are generated by random processes like Brownian motion. Random fractals aregenerated by stochastic processes rather than deterministic ones like the Mandelbrot set. Fractals ofthis kind typically show very little self similarity. [17]

2.4.4 Chaotic Attractors

While really a topic from chaos theory, chaotic attractors do show fractal behaviour and there areinteresting and sometimes completely unexpected links between fractal geometry and chaos theory.[25,2]

Chaotic or strange attractors come from the solutions to differential equations that are very sen-sitive to initial conditions. [4]An interesting example of this is the logistic map, which shares some

3The numbers are nearly always in although fractals do exist that have a complex power in them.

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interesting geometric features with the Mandelbrot set, and is defined by: xn+1= r xn(1 xn)wherex is the population at some time. This equation is the solution of differential equations about pop-ulation growth dependent on rate of reproduction and other factors. When varying the constantrwhich is the factor which scales for reproduction rate, at first an increasing rwill result in a higherpopulation as expected. However asrincreases, the population will oscillate between two levels, and

then four, and then eight, and so on. For r> 3.57 the population will suddenly become chaotic4. Agraph ofragainst initial population will exhibit fractal behaviour.

The logistic map is not an isolated example; there are many other chaotic phenomenon which willshow fractal behaviour when they are plotted. Once again, this shows that the nature of fractals ismore than theoretical; they have plenty of uses in the real world. [2,3, 4]

3 The History of Fractal Geometry

The term Fractal was originally coined by Benoit Mandelbrot in his original paper on fractalgeometry[17], however the history of these objects starts a long time before Mandelbrot wrote thatessay, with the invention of set theory and the ideas surrounding that.

3.1 Set Theory

While the inventors of set theory would have no idea that their language would be used to describefractal geometry (and indeed for many other applications, set theory is ubiquitous throughout math-ematics and science) they would form some of the first objects that could be regarded as fractals, andgive the language in which all of the mathematics of fractals would be written.

Set theory was founded in a single paper by Georg Cantor in 1874, entitled "On a CharacteristicProperty of All Real Algebraic Numbers" and in this one paper, Cantor spawned much of modernmathematics.[21]The paper considered numbers which would be solutions to an equation such as:

aixn

+ai1xn

1

+ai2xn

2

+ +a1x +a0= 0Cantor shows that there are the same number of roots to an equation such as this as there are

natural numbers, and his contempories would prove that numbers such as andecould never be theroots of an equation such as this. Cantor also went on to prove that there are more real numbers thannatural numbers, with his famous diagonal argument, and that n is of the same infinite size as .

However, it is the some of the objects that Cantor would imagine as part of his proofs that aremost interesting when looking at set theory from the point of view of fractal geometry.

While Cantor was considering the implications of set theory, he would come up with one of thevery first fractal objects: the cantor ternary set. [21,6]

3.1.1 The Cantor Ternary SetThe definition of the cantor ternary set5 is very simple: take a unit interval,[0,1], and remove the

middle third. Now repeat for the two remaining pieces. Repeat ad infinitum, and you have the set.

4Although, abover> 3.57 there are a few isolated values which will give none chaotic behaviour.5Cantor himself did not imagine or discuss the ternary set for anything more than a passing remark, more an abstracted

version of it for any repeated removal of a part of the interval. The ternary set is a more modern way of looking at Cantorsidea.

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Figure 1: The Cantor Ternary Set

Interestingly, if one considers the length of the set that has been taken away, then a surprising

result will be found. The length that has been removed at each section is equal to2n

3n+1wherenis the

number of iterations. If we sum this between zero and infinity 6 then one will obtain the following:

n=0

2n

3n+1=

1

3

1

1 23

= 1

This is interesting because it implies that even after removing an infinite number of thirds of theinterval[0,1]the length of the line segment is still the same! In my opinion this shows how counterintuitive geometries such as these can be, as this implies there are no members of the set, and yet it iseasy to see that there are.

The set shows self similar behaviour like all fractals, as any part of the set can be considered anotherenlarged and then translated. In many ways this is a prototypical fractal, the first kind which wouldhelp to establish fractal geometry as its own discipline. [6]

3.2 The Weierstrass Function

One of Cantors contemporaries and supporters was a mathematician called Karl Weierstrass.Weierstrass was the mathematician who proved that was a none algebraic number. While the revolu-tion of set theory was occurring, calculus was being formalized into the rigorous study called analysis.

Weierstrass originally published his function to challenge the notion that all functions that arecontinuous are differentiable. The function is:

f(x) =

n=0

an cos(bnx)

Where 0 1 +3

2.

The function is nowhere differentiable, as much like a fractal it has infinite detail when graphed.This means it is impossible to draw a tangent line at any point, and therefore meaning it is nowhere

differentiable.The Weierstrass function is another proto-fractal, although it is closer to what we would recognise

today as a fractal than the cantor set. [23]

3.3 The Koch Snowflake

The Kock snowflake is a curve that was constructed by Helge von Koch is another one of theproto-fractals, and was constructed due to Kochs dissatisfaction with Weierstrasss function to showthat not all functions are continuous. In contrast to Weierstrass, Kochs idea is very simple indeed.[26]

6This is possible because it forms a geometric sequence with first term1

3

and common ratio2

3

.

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Take an equilateral triangle. On the three faces of the triangle, remove the middle third of each of the lines. Replace with

another two lines to form an equilateral triangle with the part that has been removed.

Repeat for all line segments.

Figure 2: The Koch snowflake for the first 6 iterations.

The Koch snowflake shares the typical features of a fractal, such as an infinite perimeter enclosinga finite area. It it easy to see why this is the case. After each iteration the perimeter of the shapewill have increased by 4

3and therefore aftern iterations the perimeter will be( 4

3)n , and as n tends to

infinity limn

4

3

n

=

.

To calculate the area[1]inside the snowflake, one has to consider the initial area of the triangle,and the area added after each iteration. If we let a be the side length of the equilateral triangle then

the initial area is:

3a2

4. After each iteration the side length of the triangles that are added decreases

by 13

. For the first iteration, 3 triangles are added. After this, 3 4k1 triangles are added on, wherekis the number of iterations. Therefore, the area of triangles that we add on at iteration k will be equalto:

3 4k1

3

4

a

3k

2=

3

4a2

3 4k19k

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If we sum this from the 0 iteration (in this case just the first area) to the iterationnthen we will obtain:

3

4a2

1 +n

k=1

3 4k19k

If we now allown then the area of the koch snowflake will be obtained 7 .

limn

3

4a2

1 +n

k=1

3 4k19k

=

2

3

5a2

This shows that somewhat counter intuitively that an finite area can be contained within an infi-nite perimeter.[26,1,17]

3.4 The Sierpinski Triangle

Figure 3: The first 6 iterations of the Sierpinski triangle

A few years after Koch had pro-

duced his snowflake, a similar frac-tal would be produced by a Pol-ish mathematican called Wacaw Sier-pinski. Where Koch had added trian-gles, Sierpinski would take them away.If one considers an equilateral triangle,and removes a triangle from the centresuch that there are three remaining tri-angles of the same size. Now repeatthis for the remaining triangles. Con-tinue to infinity.

The Sierpinski triangle is anotherinteresting fractal, and indeed, it isa family of similar fractals, includingthe Sierpinski Carpet. As per usualwith fractals, it shows self similar be-haviour, it being a copy of itself translated and enlarged. [24,25,5,22]

3.5 The Julia and Fatou set

The Julia and Fatou sets are the basis of the Mandelbrot set that we will later explore, howeverthey are fractals themselves. First we must consider what the Fatou set is.

The Fatou and Julia sets are ideas from dynamic systems and chaos research, and indeed, fractalsand chaos do go hand in hand. A Fatou set is a set such that some function fdoes not exceed somevalue and go off to infinity when repeatedly iterated in the complex plane 8 or does go off to infitity,but has well defined propertises as it does for some complex number z . Symbolically: f :. Thefunction fmust be holomorphic 9 and generally will be a polynomial. [7]

The Julia set is the compliment of the Fatou set, meaning that it is all of the points not in the Fatouset which are in the complex plane, .

7Another geometric sequence, this time with common ratio 49

.8It does not have to be the complex plane, but all well known fractals are in the complex plane, so we will use it for our

definition.9Essentially this means infinitely differentiable.

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Commonly, the images that one will see of the Julia set are so called "Filled" Julia sets. These arewhen the interior of Julia sets are coloured to show their position with greater clarity to the reader.Symbolically this is:

K(f) = {z :f( k)(z)ask }The filled Julia set only exists for a fwhich is polynomial as for a polynomial fthe Julia set will

be the boundary of those that do go off to infinity. This is why a Filled Julia set is useful, as it showsthe section of the plane that will remain finite when iterated infinitely.

It is worth also considering the difference between a connected and unconnected Julia set. Con-nected and unconnected have the meanings you would expect them to have, when a connected set isplotted, there is no point where there is a discontinuity in the set. That is, the set is whole, and to usean analogy you could run your finger along the graph of the set without having to lift it from the page.The unconnected set is the opposite, there are breaks in the graphs of the set. This will particularlycome in useful for the Mandelbrot set, as we will now see. [7,14,15]

3.6 Benoit Mandelbrot, his set, and the essay that started it all

Benoit Mandelbrot is widely considered the father of Fractal Geometry by many, and while othermathematicians have made great contributions in the field, no one else has brought fractals to theattention of the world quite like Mandelbrot did.

While working at IBM, Mandelbrot was studying the so called parameter plane of the connectedJulia sets, for a quadratic f. That is, all quadratic Julia sets can be represented as f :zz 2 + cwherez is a complex number, and c is a complex constant. Mandelbrot was experimenting with the valuesofcsuch that when z = 0 + 0i is iterated infinitely it does not escape to infinity.

When the values of c are plotted that make the Julia sets connected, and do not escape to infinitywhen iterated, you obtain the Mandelbrot set. As iconic as it is, I would like to include a picture of it.

Figure 4: The Mandelbrot Set

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We will discuss the colouring and ways the computer plots the set and fractals in general later inthe essay.

Mandelbrot first introduced his idea of a fractal in his paper "How Long Is the Coast of Britain?Statistical Self-Similarity and Fractional Dimension" published in 1967. In it, he discusses the paradoxthat the length of coastlines will increase as the length of the measuring device decreases, and defines

the dimension of the coastline in a way that allows it to be fractional, and related to the measuringdevice being used. It would be in 1975 that Mandelbrot would coin the term fractal for objects whosHausdorff dimension was fractional, in his paper "Les objets fractals, forme, hasard et dimension"which translates to "Fractals: Form, Chance and Dimension".

Mandelbrot would finally get all of his ideas together in his book "The Fractal Geometry of Na-ture" and it was this book that brought fractals into the mathematical and scientific consciousness,and silenced critics who had believed that fractal geometry was without applications in the real worldand was just an artifact of mathematics and good programming.

While the developments in fractal geometry were taking place, similar developments in chaostheory were taking place, and the two ideas would feed into one another, sharing similar roots math-ematically. Between the ideas of chaos theory and fractal geometry, many believe that the idea of a

clockwork universe 10 was brought down entirely. Indeed, as mentioned earlier there are surprisinglinks between the Mandelbrot set and an idea from chaotic systems called the logistic map, and objectscalled chaotic attractors quite commonly have fractal features. [17, 24, 27, 18]

3.7 After Mandelbrot

Once Mandelbrot had established the field of fractal geometry it would only be a matter of timebefore others would start to use his ideas for other uses, and use them they did. From art to theoreticalphysics, fractals have been used all over. These uses are what the rest of the document will deal with.

4 Generating images of Fractals

The generation of the images of fractals has been revolutionised by the advent of computers. Plot-ting images of fractals lend themselves very well to computers, as what is required is simple calculationsrepeated hundreds or thousands of times, for hundreds or thousands of test points and computers areperfect for such calculations.

Indeed, for many years, the beautiful images of the objects that mathematicians were imaginingwere out of reach for them, due to not being able to perform the necessary calculations to view theimages they theorized about.

The simplest way of plotting fractals is an escape time algorithm. Since we cannot actually iteratea function to infinity, we instead must develop tests to show when the function is going to escape toinfinity and when it will stay bounded. For the Mandelbrot set, this is easy as it has been proved that

once|z| 2 after some number of iterations, then the point will always escape to infinity. Otherfractals will have similar tests like this. Once a point has been determined to be either in the set or outof it, the point can be plotted appropriately.

In order to colour the edge of the fractal one has to consider how quickly a value reached the escapevalue. Usually, black is the colour that is used for values which do not escape at all and graduallylighter colours imply that after fewer iterations the point escaped to infinity, with white meaning thatthe point immediately passed the test for divergence.

10A universe where a knowledge of all of the initial states of all particles and exact knowledge of the laws of physics wouldallow you to calculate the evolution of the universe from that point onwards exactly.

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An interesting optimisation that can be used for the Mandelbrot set is to substitute for z =x +yiand notice what happens when this is squared.

z=x +yi

z2 =x 2

y2 + 2xyi

Now when calculating the next value ofzin the iteration for the Mandelbrot set one can use theseformulas, withc=x0 +y0i .

xn+1 = Re(z2 + c) = x2

ny2

n+x0

yn+1 = Im(z2 + c) = 2xnyn+y0

Another more computationally challenging algorithm to colour a fractal is the Normalized Iter-ation Count algorithm. The escape time algorithm is quick for colouring, but it produces bands ofcolours, which can be quite unattractive. The Normalize Iteration Count algorithm is more aesthet-ically pleasing. The algorithm works by assigning a number to each value ofz after however manyiterations are being calculated. If this number isv then the formula will be:

v= n logPlog2 |zn |Wheren is the number of iterations, Pis the power that is used in the formula for the fractal, for

the Mandelbrot set this is 2, andzn is the value ofz aftern iterations. Once computing this measurethe programmer can then assign a more flexible array of colours to the fractal, and therefore producea more aesthetically pleasing image as a result. [17, 25,24,27]

5 Fractals in Art

Fractal artwork has to some extent, always been around. Humans have always been attracted toself-similar shapes, as they are some of the most beautiful shapes found in nature. Of late though,fractals have began to be used in other more interesting places, and found in places that they wouldnot have been expected to be in.

One of the most interesting places that fractal geometry has been found is in aerial photographs ofAfrican villages. In order to fit with their customs they will build their villages into shapes which willfollow fractal geometry, which is a particularly amazing idea. The ideas used are beautiful, and the vil-lages they build from the air are also particularly beautiful. These patterns are different dependent onwhere you go into Africa, and seem to occur all over the content, which shows the unique attractionthat humans have towards self-similar geometry.[11]

Another interesting place that fractal geometry has been found is in the artwork of Jason Pollock.After mathematically analyzing the patterns that are found in his paintings, mathematicians foundthat they have features in common with fractal geometry. It has been suggested that this is the reasonthat Pollocks seemingly chaotic paintings still have a visual appeal to people, as underneath they havea self-similarity on several scales, much like in nature.[8]

Recently, the increasing availability of vast amounts of processing power to individuals has meantthat fractals have moved from the province of merely academic discussion into something that anyonecan play with after downloading a small program from the Internet. My own initial experiences withfractals were like this.

This increase in computing power has lead to a wide variety of fractal images being freely availableon the Internet, and plenty of resources there for people to find out about fractal geometry. The greatnumber of free programs to visualise fractals beautifully, and the widely available example implemen-tations of fractals in various programming languages, means that almost anyone can play with theseobjects, which is a great thing.

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6 Fractals in Science and Nature

Since nature is chaotic and self-similar, it was only a matter of time before fractal geometry beganto be applied to situations in nature. Fractals scale invariance means they are useful in all sorts ofplaces, such as weather analysis. Here I hope to only show a few of the brilliant examples of fractals

in the real world.

6.1 Fractals in Weather analysis

The weather is the classic example of a computationally hard, complex chaotic problem. Theequations that govern how the weather evolves over time are a series of difficult partial differentialequations with chaotic solutions. However, recently, mathematicians have began to consider howthese problems could be looked at using fractal geometry.

It is easy to see how the weather shows self-similar behaviour, and how smaller effects can easilyfeed into larger ones and currents look similar on large scales as they do on small ones. These arenot just theoretical predictions of what the weather will look like either, actual weather data has beenanalysed and shown to have fractal behaviour when looked at.

Indeed, both rainfall and clouds already have quite successful fractal models for their behaviour,and so modeling the climate using fractal models certainly does not seem like an impossibility. [19]

6.2 Fractals in Theoretical Physics

The very first type of fractal which physicists considered was that of Brownian motion. Therandom motion of a particle held in a fluid will form a fractal with statistical self similarity.

Another interesting use of fractal geometry is fractal cosmology. While it is not considered a majortopic within cosmology, the structure of galaxies and other objects can be considered fractal, and theirproperties studied. This is a relatively new idea, and therefore has not as of yet produced any majorlylarge results, but the potential is there for it to produce some interesting ideas.

Recent observations also suggest that the overall structure of the universe can be considered fractalin some respects, and the consequences of this are still uncertain.

Mandelbrot in his book "The Fractal Geometry of Nature" discusses the use of Fractal geometryin lattice physics, that is the physics of how atoms align themselves within solids. He argues that manyof the structures found in already well understood crystals exhibit fractal behaviour and that analysingthese properties might lead to a new understanding of how condensed matter acts. Once again, thisis due to the fact that small changes in the arrangement of the lattice structure of the solid can causemuch larger changes to the overall structure.

Finally, fractal structures have been found when observing the magnetic phase-transitions of cer-tain materials, that is the temperatures at which they become magnetic. By considering the mathe-matics of these transitions, two physicists Yang and Lee would produce another set of fractals. These

fractals are just as beautiful as the other examples given so far, and indeed, one of them contains ele-ments of the Mandelbrot set, which is a very interesting and deep coincidence. [17,9]

6.3 Fractals in Biological systems

One of the most obvious examples of fractals within nature is Barnsleys fern, although this is notan isolated example, as plenty of biological systems show fractal behaviour.

An interesting example of this is the growth of bacteria, which spreads out in a pattern that isobviously fractal in nature. These shapes have been proved to be fractal in nature, and while theyseem to have little application to biologists, it is an interesting pattern to be aware of.

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In a similar fashion to this, the lungs can be considered fractal, once again with self-similaritydown to the small alveoli. When measuring these with smaller and smaller measuring devices thesurface area of the alveoli increases dramatically, much like the coastlines which Mandelbrot discusses.The lungs themselves can be visualised as a fractal canopy, which is a form of iterated function system.[9]

6.4 A few other examples

Doctors are considering using fractals to examine heart traces, as there is some evidence thatexamining the fractal nature of the heart rate allows information to be extracted that would nothave otherwise been obtainable.[10]

Geologists making models of water moving through soil are exploring using mathematical mod-els for the path of water through the soil.

Video games designers and film makers have used random fractals and iterated function systemsto model landscapes, as these produce realistic back drops quickly and easily. [25]

7 Conclusions

Fractal geometry is still an emerging topic with much left to offer to mathematics and scientificknowledge in general. In this essay I have barely touched the surface of the mathematics and uses offractals. Fractals are beautiful objects, with remarkable properties that will allow them to continue tobe some of the most important objects in mathematics. I predict that as fractals gain more exposurewithin the scientific community, they will continue to be used in ever more diverse models for physicalphenomena.

Fractals are now within the general consciousness of our culture, and will continue to inspire bothart and science for many years to come. Long may these enigmatic and beautiful objects have the place

they deserve.

References

[1] Area of the koch snowflake. URL:http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/area.htm.

[2] Attractors. URL:http://en.wikipedia.org/wiki/Attractor.

[3] Attractors on scholarpedia. URL:http://www.scholarpedia.org/article/Attractor.

[4] Attractors on wolfram mathworld. URL:http://mathworld.wolfram.com/Attractor.html.

[5] Michael Barnsley.Superfractals. Cambridge University Press, 2006.[6] Cantor Set. URL:http://en.wikipedia.org/wiki/Cantor_Set.

[7] Conversations with Gaetan Bission regarding the Hausdorff dimension and Julia and Fatou Sets.URL:http://www.win.tue.nl/~gbisson/en/.

[8] Discover magazine on Jackson Pollocks Paintings with fractals. URL: http : / /discovermagazine.com/2001/nov/featpollock.

[9] Fractal Applications. URL: http://library.thinkquest.org/26242/full/ap/ap.html.

[10] Fractal heart beat analysis. URL: http://www.physionet.org/tutorials/fmnc/node10.html.

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http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/area.htmhttp://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/area.htmhttp://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/area.htmhttp://en.wikipedia.org/wiki/Attractorhttp://en.wikipedia.org/wiki/Attractorhttp://www.scholarpedia.org/article/Attractorhttp://www.scholarpedia.org/article/Attractorhttp://mathworld.wolfram.com/Attractor.htmlhttp://en.wikipedia.org/wiki/Cantor_Sethttp://en.wikipedia.org/wiki/Cantor_Sethttp://www.win.tue.nl/~gbisson/en/http://discovermagazine.com/2001/nov/featpollockhttp://discovermagazine.com/2001/nov/featpollockhttp://library.thinkquest.org/26242/full/ap/ap.htmlhttp://library.thinkquest.org/26242/full/ap/ap.htmlhttp://www.physionet.org/tutorials/fmnc/node10.htmlhttp://www.physionet.org/tutorials/fmnc/node10.htmlhttp://www.physionet.org/tutorials/fmnc/node10.htmlhttp://www.physionet.org/tutorials/fmnc/node10.htmlhttp://www.physionet.org/tutorials/fmnc/node10.htmlhttp://library.thinkquest.org/26242/full/ap/ap.htmlhttp://discovermagazine.com/2001/nov/featpollockhttp://discovermagazine.com/2001/nov/featpollockhttp://www.win.tue.nl/~gbisson/en/http://en.wikipedia.org/wiki/Cantor_Sethttp://mathworld.wolfram.com/Attractor.htmlhttp://www.scholarpedia.org/article/Attractorhttp://en.wikipedia.org/wiki/Attractorhttp://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/area.htmhttp://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/area.htm


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[11] Fractals in africa from TED talks. URL: http://www.ted.com/talks/ron_eglash_on_african_fractals.html.

[12] Hausdorff Dimension. URL:http://en.wikipedia.org/wiki/Hausdorff_dimension.

[13] Hausdorff Dimension Wolfram Mathworld. URL: http : / / mathworld . wolfram . com /HausdorffDimension.html

.[14] Julia Set. URL:http://en.wikipedia.org/wiki/Julia_set.

[15] Julia Set Wolfram Mathworld. URL:http://mathworld.wolfram.com/JuliaSet.html.

[16] Hans Lauwerier.Fractals Images of Chaos. Penguin Books, 1991.

[17] Benoit Mandelbrot.The Fractal Geometry of Nature. W. H. Freeman, 1977.

[18] Mandelbrot set on wolfram mathworld. URL: http : / / mathworld . wolfram . com /MandelbrotSet.html.

[19] NewScientist: Tomorrows weather: Cloudy, with a chance of fractals. URL: http://www.newscientist. com/ article / mg20427335 . 600- tomorrows- weather - cloudy- with - a -

chance-of-fractals.html.[20] Self-Similarity. From MathWorld. URL: http : / / mathworld . wolfram . com / Self -

Similarity.html.

[21] Set theory. URL:http://en.wikipedia.org/wiki/Set_theory.

[22] Sierpinski Triangle. URL:http://en.wikipedia.org/wiki/Sierpinski_triangle.

[23] Weierstrass Function. URL:http://en.wikipedia.org/wiki/Weierstrass_function.

[24] Wikibook on Fractals. URL:http://en.wikibooks.org/wiki/Fractals.

[25] Wikipedia: Fractals. URL:http://en.wikipedia.org/wiki/Fractal.

[26] Wikipedia on the koch snowflake. URL:http://en.wikipedia.org/wiki/Koch_snowflake.

[27] Wikipeida on the Mandelbrot set. URL:http://en.wikipedia.org/wiki/Mandelbrot_set.

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