I.B. Extended Essay :
Complex Simplicity, An Investigation into Fractal Geometry

I. Introduction.

Mathematics has become the most essential tool of our modern lives. It is called, and not in vane, ‘The Queen of the Sciences’ , the science from which almost every piece of knowledge comes from and to which unquestionably all disciplines refer to with the objective of forming a solid base to their theories and laws. But this is partly so, because humanity has made it this way, because we have created our world around mathematical concepts. At the times of its origins, mathematics were intended to be a model which could explain the everyday events but as the model started to become more complicated and some people started to appreciate and develop the fascinating aspects of pure math, the original model grew apart of the natural world. However, today we stand before what could be the convergence of mathematics and nature, today we stand before fractals.

Fractals are graphical representations of mathematical equations, equations that are often quite simple. Fractals have some basic characteristics:

•  They are self-similar, i.e. a magnification of the fractal looks like the whole which means they lack scale (see Figure 1 below how each branch is exactly like the entire fern).

• They can have dimensions outside the normal Euclidean ones of integers 1,2 and 3. Fractal images can have dimensions of 1.5, 2.3333... or log 5/log 2.

 The main difference between fractals and normal graphs such as that of a sine wave or an y=x² parabola is that these images are based on a concept known as iteration. Iteration involves using the result of an equation as a starting parameter to calculate the same equation again, in other words, it is a feedback loop where one result depends directly on the previous one. Although this concept was known a long time ago, it was not until a few decades that it was considered properly. This is because of the great amount of calculations required to gather enough results to see the astonishing patterns. So, with the arrival of computers able to calculate thousands or even millions of operations per second, investigation into iteration was made possible. It was at this time when the French-Polish mathematician Benoît Mandelbrot assembled the work of many mathematicians like Gaston Julia, Henri Pointcaré and also with his own important contributions to give birth to fractal geometry : "I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means ‘to break:’ to create irregular fragments." Mandelbrot claims that the fractal images represent the patterns of nature and although this is only a theory, the resemblance between fractal images and natural objects is undeniable (see Figure 1 below).

As I said before, Mandelbrot couldn’t have done what he did if it weren’t because of computers ; it was this magnificent tool that his predecessors lacked and without which, although they were brilliant they couldn’t conclude their work. Being no exception, I shall turn to the computer as well to analyze graphically all data and to generate the images that will appear in the course of this work. In this essay, I will explain the concept of fractal dimension introduced above, investigate the Sierpinski triangle and the Mandelbrot and Julia sets and finally attempt to generate a fractal using a (rather simple) complex quadratic equation. The main software (programs) that I will use are Fractint v19.2, which is an excellent fractal generating and manipulating program and QuickBASIC 4.5 which is a programming language with which I will attempt to generate myself some fractal images. Unless stated otherwise, all fractals appearing in the Extended Essay are generated with the first program.

II. Fractal Dimension.

 1. How long is the coast of Britain?

 In one of his most famous papers, Mandelbrot raises this apparently simple question. After little consideration, we find out that the answer is not just a simple number but it actually depends on the scale at which the coastline is measured. As we all know, a natural coastline is far from being a straight line ; it is composed of bays and headlands and at the same time, these are composed of smaller bays and headlands. To measure the length of these protuberances, we will take the following method: A compass with an opening of distance n is put at the beginning of the curve and is passed through it taking a step at each consecutive point n units away. At the end the number of steps (s) taken is counted and the total estimate for the length is calculated by multiplying by n, That is :

As the opening of the compass is reduced, the total effective length of the coastline increases without limit as the method each time takes into account the finest details as the cracks of each stone when n is in the order of a centimeter. Taken this into consideration, it can be said that the length is actually infinite. Therefore it would be fair to say that this curve has a higher dimension than that of a simple line though less than that of a square that fills the plane. Then how can it be stated that the coast is twice as long as each of its halves without falling into the paradox of infinity being twice or three times as big as infinity which, although it’s true, has no use for us now? To answer this question a way to describe exactly how complicated the coastline is needs to be looked at: this is the fractal dimension.

2.Self-similarity dimension.

The variation of L(n) with respect to n was studied by Lewis Fry Richardson in 1961 and he discovered that L(n) was proportional to n¹-a , a being a constant, and that this did not change with the method used to determine the length whether it is the one described earlier or one of the many others. The value of a depends on the chosen coastline and within the same coastline different parts will render different values for a . Thus, this leads to the thought that a , being inherent to the specific coastline, could be a measure of how complicated it is. To simplify the calculations involved herein, a simplified model of the coastline will be used: the Von Koch curve. The way to produce this line shows the close connection to a coastline. The initial state, a line segment, is divided into three equal parts and the middle section is replaced with two lines forming a bottomless equilateral triangle. This protuberance is the equivalent to the headland of a coastline. This leaves us with four segments to which the procedure is applied over and over thus leaving an infinitely complicated curve (Figure 2). Curiously, this curve follows Richardson’s law but in this case, dimension a (D) can be calculated exactly.

As it can easily be seen, the dimension D does not vary with n and is therefore a measure that can be used to compare the level of complexity.

This dimension has a close relationship to the way in which the curve is self-similar. If we take a close look at the first iteration (Stage 1) we see that it has four times the amount of segments as the initial line (Stage 0) which need to magnified times three in order to make them identical to the initial one. Exactly the same can be said of Stage 2 to Stage 1 and any other Stage x to Stage x-1. As we can see, the value for the dimension (log 4/log 3) can be calculated as log(parts)/log(scale). This concept can be applied to (or deduced from) any geometrical shape with self-similarity and this includes the normal Euclidean figures as the line, square and cube with normal dimensions 1,2 and 3 respectively. A line can be divided into two parts and each of these parts needs to be magnified times 2 to be the same as the original, thus D = log 2/log 2 = 1. Equally a square divided into four smaller ones each needed to be magnified times 2 to get the original one, D = log 4/log 2 = 2 and the cube, log 8/log 2 = 3 (see Figure 3).

Now that the method for determining the fractal dimension of an object is generalized it can be applied to any fractal. To show this, I will calculate the fractal dimension of Figure 4 below, which is a fractal curve very similar to the Koch curve. It is obtained by dividing the original line (Stage 0) into five segments and replacing the middle three for four of the same size as shown leaving a figure with six parts each five times smaller than the original segment.

The dimension for this curve is slightly smaller than the one for the Koch curve, however both curves have infinite length. Therefore this measure is perfectly adequate to compare the complexity of a fractal.

III. The Sierpinski Triangle.

1. Construction.

The Sierpinski triangle is one of the most simple and at the same time one of the most fascinating fractals. This is partly because of the numerous ways in which it can be formed. One of the easiest and most intuitive way is as follows : Take an equilateral triangle, cut all its sides in two and join this midpoints to divide the triangle into four equal pieces and remove the central piece. Repeat this procedure with the remaining triangles to infinity. This method is shown in Figure 5 below.

Using this method of construction, we can see that the perimeter increases with each stage, but the area diminishes dramatically ; in fact, the Sierpinski triangle has no area and its perimeter is infinite. A proof for this is seen in the table below: in the formula for the perimeter, 3^N+1 will increase at a higher rate than 2^N therefore the perimeter will tend to infinity as N does and in the formula of the area, (5/16)^1/2 is a constant and 2^-2N will decrease at a higher rate than the increase rate of 3^N making the area tend to zero as N increases. (See table below)

Another method to produce this amazing figure is probably more amazing that the figure itself. It’s based on Pascal’s triangle. Pascal’s triangle is used in probability theory, number theory and polynomial expansion and, curiously enough, it can help in the creation of Sierpinski’s triangle. The procedure is as simple as this: Imagine each integer as being on a small box. If the number is odd, the box is painted black, if it is even it’s left unfilled. This incredibly simple procedure will produce the triangle as shown in Figure 6 below.

This is only a piece of the triangle, of course, but it’s know that Pascal’s triangle is infinite and this pattern continues all the way down the triangle. Knowing that two odd or even numbers add up to an even number while a combination will result on an odd number proves that the pattern continues indefinitely. The main difference between this representation of the triangle and the normal one is that the normal one has definite corners and can be zoomed infinitely while this one cannot be zoomed but extends infinitely creating larger and larger triangles.

A third method to construct the Sierpinski triangle is by an iterated function system or IFS. An IFS consists on two basic things, transformations and probabilities: the transformations, usually expressed as matrixes, are two: one transformational matrix (2´ 2) and one translational matrix (2´ 1). These matrixes are applied to the coordinates of an initial arbitrary point and the result plotted and fed back to the equation. IFSs consist of more than one group of matrixes and the probabilities are used to decide which group of matrixes will be applied to the point. In the case of the Sierpinski triangle there are three matrixes, each with 1/3 probability. The matrixes are as follow:

Choosing any point on the plane and applying the matrixes selected at random will result on the triangle as in Figure 7 below. One of the virtues of this method of construction is that it consists of simple procedures and it has no stages ; any point plotted by the IFS will definitely be part of the triangle. Note that the transformation is the same for all three groups of matrixes and indicate the midpoint while the translations are the coordinates of the corners of the original triangle. This means that the Sierpinsky triangle can exist as any kind of triangle, not only equilateral ; just changing the values in the translational matrixes will render a Sierpinski triangle of any shape and size.

2. Dimension and variations.

As discussed before, the Sierpinski triangle has an infinite perimeter and no area. This suggests that its dimension should lie between that of a line (1) and that of a plane (2). Complying with one of the most important characteristics of fractals, this figure has a noticeable self-similarity. If we look at any of the figures of the triangle above, we can see that each of the three major sub-triangles can be magnified and would look identical to the whole. Knowing this the method of self similarity described in Part I of this essay can be used to calculate the dimension of the Sierpinsky triangle:

                              Referring to Figure 5 (page 9) :

This shows that the Sierpinski triangle is more complicated than the Von Koch curve and the other models of coastlines. This makes sense considering that those were made from lines and adding while this was made subtracting from a triangle.

As said before, the Sierpinski triangle is one of the simplest fractals and there are literally thousands of variations of the original method used to generate Sierpinski-like fractals. Here are some examples of these and a calculation of their dimensions. The first is originated by changing the original triangle for a square and the second is done by applying the construction method to a tetrahedron to obtain a 3D-aspect Sierpinski. This last is a perfect example of a fractal object with integer dimension.

IV. The Mandelbrot and Julia Sets.

1.Construction.

The Mandelbrot and Julia sets belong to a different kind of fractal than those seen up to this moment in this essay. The reason for this is that they are graphed using a raster method. This means that each point in the image is evaluated independently according to a specified method, therefore the points used are chosen from the complex plane at regular real and imaginary intervals leaving it as a sort of tidy grid. However this is done at a high resolution so the eye doesn’t notice the little squares. For the Mandelbrot set (M-set) and Julia set (J-set), the equation used is the following:

where z and c are complex variables. For the time being however, we’ll consider c to be constant. For the M-set, each point in the plane represents a value for this constant c. The equation is iterated starting with z = 0. If after a given number of iterations z grows to infinity, the point for c is outside the set ; if, however, z remains within certain values and never boosts out, the point is inside the set. In brief, the M-set is the set of values of c for which the above equation stabilizes upon iteration. Originally the Mandelbrot fractal was of one color showing if the point was inside or outside the set but the points outside the set can be colored according to how fast (i.e. on which iteration) it shoots to infinity. This makes the image incredibly beautiful (see Figure 10 below).

This figure is the complete M-set ; it exists fully within a small region close to the origin. In the diagram the corners are (-2.5,-1.5) bottom left and (1.5,1.5) top right. This fractal is also self-similar, although this feature is not as obvious as in the ones seen before. The following diagram shows a blowup of the tiny white spot seen at the middle-left of the previous one:

Note that the figure is identical to the whole (The outside colors are different but remember they are not really part of the set).

The Julia set is constructed in the same way as the M-set. The difference is that for a Julia set, the constant c remains with the same value not only during the iteration of each point but throughout the whole image. The real and imaginary values of each point in the complex plane to be iterated with the equation represent the initial value of z (which was constant at 0 for the M-set). This means that the Julia set is in the dynamical space (z-plane) while the Mandelbrot set is on the parameter plane (c-plane). The image below (Figure 12) shows the Julia set for c = .3+.6i:

2. Mandel/Julia relationship.

As it can be seen, the M-set and J-set are strongly bound together as the only difference between them is the value that varies according to the plane and the one that remains constant. As every point on the Mandelbrot set represents a value of c, and the Julia set uses only one value of c throughout the image, it follows that for every point in the Mandelbrot set there is an entire Julia set which is directly connected to it. If the point of the Mandelbrot fractal used to generate the J-set is inside the set, then the J-set will be connected, otherwise, it will be just a dust. This interrelation between both sets is very well known and documented. It follows logically that if the Julia set varies a value that is kept constant in the M-set, for each point in the Julia set there is an entire Mandelbrot set. This characteristic is not as documented as the latter one, probably because the M-sets that have initial values of z differing from the traditional 0 are not even close to be as pleasing as the original set. The initial value of z in such a M-set is called the perturbation of z. Below is a figure of a non-standard M-set for which the perturbation of z = .5 + .5i:

After examining these relationships I got to the conclusion that not only were these two sets very closely related but that they were parts of the same fractal. The Julia-Mandelbrot fractal is a four-dimensional figure in terms of both complex variables z and c. Both the M-sets and the J-sets are bidimensional slices of this figure, both sets are slices parallel to two of the four axis (real and imaginary component of the complex variables), this idea would be the same as the face of a cube. In the same way that we can get 2D slices of a cube that are not parallel to the faces we can get other kinds of 2D slices from this 4D fractal. At least four more sets which are parallel to two axes (the combination of real and imaginary parts of z and c) and an infinity of more complicated slanted ones. It is one of the parallel sets that I will attempt to graph with a home-made program in QuickBASIC.

3. Home made fractal.

The fractal I will generate is a slice of the four-dimensional figure discussed before. The method is the same as for any Mandelbrot set and any Julia set but instead of varying either c or z, I’ll vary one component of each while keeping constant the other two. In order to do this (and because the computer doesn’t handle complex numbers), I need to split the variables z and c into zr , zi and cr and ci. To generate the following images, I froze cr (real part of c) and zi (imaginary part of z) at a value of 0 and varied zr on the x-axis and ci on the y-axis.

The figure above has as corners (-4,-3) and (4,3). The figure below is a zoom of the upper middle part with corners (-0.4,0.5) and (0.4,1.1).

This last figure is a zoom from the little antenna-like part at the middle-right of the 1^st zoom with corners (0.155,0.81125) and (0.225, 0.86375).

With these images of the Zr,Ci plane I can see that the program worked well. A copy of it can be found in Appendix B.

V. Conclusion.

After having completed this investigation I can arrive to various conclusions. In the first place, I have learned about the nature of fractal objects and the key points that differentiate them from Euclidean objects such as their infinite complexity and self-similarity. I could arrive at a method of comparing fractal curves according to the degree of infinity of their perimeters or the degree of complexity, the concept of fractal dimension and how it is calculated. In the section involving fractal dimension, I used the Koch curve as a model of a coastline and unfortunately, this amazing curve couldn’t be treated at a deeper level.

In regards to the Sierpinski triangle, I arrived at a general formula for its perimeter and area at any stage of its geometrical construction, I could effectively calculate its dimension and cover various ways of generating the fractal. I could also use this figure as a general pattern that opened the doors to a whole universe of fractals such as those in Figures 8 and 9 which obviously couldn’t be thoroughly analyzed within the limits of the essay.

With the Mandelbrot and Julia sets, I could explain the raster type of fractal as well as establish and demonstrate the tight link between the two sets. On top of this, I developed a theory stating that these two sets are slices of a four-dimensional fractal and although the images generated by the QuickBASIC program appear to support this theory, I couldn’t find enough documentation to eliminate all doubt. This is an area into which I would have liked to concentrate but it wasn’t possible due to lack of resources.

As a general conclusion, I can say that the study proved very useful and rendered good results.

VI. Bibliography.

Mandelbrot, Benoît. Los objetos fractales. Tusquets Editores, Barcelona (1993)

Mandelbrot, Benoît. The Fractal Geometry of Nature. W. H. Freeman and Co., New York (1977) 

Chaos Theory at http://tqd.advanced.org/3493/noframes/chaos.html

Fractals and Fractal Geometry at http://tqd.advanced.org/3493/noframes/fractal.html

Fractals and Chaos Theory at http://passion.roc.servtech.com/chaos/fractal.html

Fractal Dimensions at http://www.tep.ucsd.edu/spacemath 2/spacemath/fracdimension.doc

Chaos Homepage at http://www.students.uiuc.edu/~ag-ho/chaos/chaos.html

Sierpinski Triangle at http://www.students.uiuc.edu/~ag-ho/chaos/sierpinski.html

Noel Giffin High Iteration Fractals with Fractint at http://spanky.triumph.ca/www/fractint/hi-iter.html

E.Stepp. Fractal FAQ at http://www.marshall.edu/~stepp/fractal-faq/faq.html

VII. Appendix A.

This is a program (original) written in Microsoft QuickBASIC 4.5 that generates the Sierpinski triangle with the IFS method as shown in Figure 7. Note however that the one in Figure 7 was generated with Fractint 19.2.
DOWNLOAD

DEFDBL A-Z
SCREEN 12
WINDOW (0, -.2)-(2, 2.5)
DO
Matrix = RND
IF Matrix <= 1# / 3# THEN
RPX = .5 * RPX
RPY = .5 * RPY
ELSEIF Matrix <= 2# / 3# THEN
RPX = .5 * RPX + 1
RPY = .5 * RPY
ELSEIF Matrix <= 1 THEN
RPX = .5 * RPX + .5
RPY = .5 * RPY + SQR(1.25)
END IF
PSET (RPX, RPY)
LOOP WHILE INKEY$ = ""

VII. Appendix B.

This is a program (original) written in Microsoft QuickBASIC 4.5 that generates the images shown in figures 14,15,16 of the Zr,Ci slice of the Mandel/Julia 4D fractal. To obtain the different zooms replace the original coordinates with the ones of the desired zooming area.
DOWNLOAD

DEFDBL A-Z
XMin = -4 ' Change the corner values here
XMax = 4 ' to achieve desired zooms as
YMin = -3 ' shown by the different figures
YMax = 3
SCREEN 12
WINDOW (XMin, YMin)-(XMax, YMax)
YStep = (YMax - YMin) / 480
XStep = (XMax - XMin) / 640
FOR Ci = YMin TO YMax STEP YStep
FOR sZr = XMin TO XMax STEP XStep
BAIL = 0
Zi = 0
Zr = sZr
FOR IT = 1 TO 50
ZrTemp = Zr * Zr - (Zi * Zi)
Zi = 2 * Zr * Zi + Ci
Zr = ZrTemp
IF Zi * Zi + Zr * Zr > 4 THEN BAIL = 1: EXIT FOR
NEXT
IF IT = 51 THEN IT = 0
PSET (sZr, Ci), IT MOD 16
NEXT
NEXT

Abstract.

Complex Simplicity is an essay about fractal geometry. This is a new branch of mathematics that differs from normal Euclidean geometry. In this essay, I have concentrated in the investigation of fractal dimension and the analysis of two kinds of fractals with their most important representatives: the Sierpinski triangle and the Mandelbrot set. Within the investigation of the Mandelbrot set there is a short section showing how a fractal is generated using any common computer emphasizing the concept that however complicated a fractal is, it’s formed by simple equations.

In the course of the investigation, the nature of fractal curves was discussed and their role as mathematical models for natural things such as a coastline. The concept of fractal dimension was introduced and explained. The Von Koch curve was used as the model for these curves and its dimension was calculated. The method was generalized for any type of fractal curve. The Sierpinski triangle was analyzed with it’s various methods of construction and these were compared. The dimension of the triangle was calculated and the results were again generalized to suit any fractal which fits the general Sierpinski model. Finally the Mandelbrot set was studied together with the Julia set. While studying these fractals, I developed a theory that links them together and information deduced from this was used to graph the home-made fractal.

The conclusions to this project were many, and so were the questions that arose from the investigation. Concerning the fractal dimension, a general method was established for all fractal curves. A general formula was developed to calculate the area and perimeter of the Sierpinski triangle at any stage and its dimension was calculated as well. The investigation on the M-Set gave good results but the theory I developed couldn’t be confirmed or refuted.