An Introduction to Fractals

An Introduction to Fractals Sarah Hardy December 10, 2018 Abstract Fractals can be defined as an infinitely complex pattern that is self-similar, that is contains replicas of itself of varying sizes, across different scales. Analyzing the definitions of a fractal unveil the ways in which these shapes differ from the average and therefore introduce interesting concepts. Fractals as they appear in nature and as they can be produced mathematically are complex patterns that often result from simple iterative equations. In this paper, we will define simple fractals and extend these ideas to create more complex forms such as random fractals and the Mandelbrot set. 1 Introduction Although the term fractal may not be universally well known, everyone has most certainly observed fractals without realizing it. From tree branches to mountain ranges, rivers to Romanesco broccoli, fractals exist in many forms throughout nature. (a) A tree [1] (b) Romanesco broccoli [2] Figure 1: Two examples of natural fractals Formally, a fractal is an infinitely complex pattern that is self-similar across many scales [3]. This self-similarity is necessary to make fractals unique and define them apart from other shapes in nature and mathematics. We can think of a self-similar object as one that contains replicas of itself of various sizes [4]. Consider one of the natural fractals listed above such as the tree branch. If you were to break off a branch from a tree, it looks like a smaller version of a tree, with limbs branching off from each other. If you were again to break off one of these branches, you would again see what looks like a smaller version of the whole tree branch. While in nature, the smaller branches you see will not be the exact same shape as the whole initial branch, the shapes that make up the branch mimic that of the overall and thus we see this idea of self-similarity. 1

Not only can fractals be found in nature, but they can be produced mathematically. Mathematical fractals can be exactly self-similar at different scales, however, as we explore mathematical fractals, we will see that this is not always the case and these imperfect examples give us a more interesting topic of exploration. 2 Defining Fractals With self-similarity in mind, we can explore how to create simple fractals to deepen our definition then expand these ideas in an attempt to model natural phenomena. 2.1 Iterative Methods The complex shapes of fractals can be created quite easily by the simple idea of iteration. An iterative function begins with an initial condition x 0 and a function f. The output of f(x 0 ) = x 1, the first iteration, is put back into the function as f(x 1 ) = x 2 for the second iteration and so on for a given number of iterations or indefinitely. We can see this applied geometrically in an example of a fractal snowflake: Figure 2: A fractal snowflake defined at iterations n=0,1,2,3 where the function is to create 4 copies of the object and place one at each corner [4]. As we can see in this example, a complex object is created by a very simple process. The resulting shape is indeed a fractal by our definition of self-similarity. Zooming in to any one section of the fractal snowflake will look exactly like the whole object. Similar methods of iteration can be used with different shapes or functions to create unique fractals. We will expand upon this idea of fractal creation via iterative functions as we explore the Mandelbrot set. 2.2 Self-Similarity Dimension Shapes are often defined by their size in various ways including length and area. If we want to instruct someone to draw a square, for example, we can ask them to draw a square of area 1 in 2 or with sides of length 1 in. Similarly, a person has an average size, measured by your height for example. However, a shape that is infinitely self-similar doesn t have the usual definition of an average size. To understand this concept, consider if you lived in a fractal world such as our fractal snowflake from Figure 2, and you were shrunk. In the real world, if you were suddenly shrunk, you would be able to tell based off the average size of the objects around you. Your dog would be bigger than you, your bed would be too tall for you to get into, and everything 2

around you would appear larger than usual. On the other hand, if you were shrunk in our fractal world, you would have no way of knowing by how much your size had changed, if at all. The same shape would still appear in scales larger and smaller than you and thus you would have no concept of how much your size had changed. As this example demonstrates, the idea of average size is not useful when defining a fractal. Thus, we must define a different form of measurement to classify fractals. Although the average size of a fractal is not a useful measurement, we find that comparing the scale of different iterations of a fractal gives insight into what makes the fractal shape unique. The self-similarity dimension defines how many small self-similar pieces of an object fit within a larger piece. To find the self-similarity dimension, D, we can solve the simple equation: (number of small copies) = (magnification factor) D. The origins of this equation can be traced back to simple known dimensions. First, consider a line segment. If you magnify this line segment by 3, you now have 3 small copies of the line segment and a magnification factor of 3, and therefore the new figure has dimension 1. Similarly, if you start with a square and magnify it by 3, the length and width of the square will be three times its original size, however you can now fit 9 of the original sized squares in the new square. Thus, the dimension is 2, as we would expect for a square. Now that we see this equation holds for basic 1 and 2 dimensional objects (and you can easily prove it holds for a cube as well), we can consider how it will work for a fractal. Once again, consider our fractal snowflake in Figure 2. Each successive iteration of the snowflake has height and width that is 3 times the size of the original object and contains 5 small copies of the shape. Thus, every element of the fractal snowflake is made up of 5 smaller elements that look like the element itself but scaled down by 3. Plugging this into our equation, we can solve 5 = 3 D to get D = 1.46. We can interpret this value to mean the fractal snowflake is somewhere between a one and a two-dimensional object. To expand our definition of a fractal, we can compare this self-similarity dimension to the topological dimension. The topological dimension of an object is the more intuitive notion of dimension where a point is 0- dimensional, a line is 1-dimensional, etc. The snowflake, for instance, has topological dimension of 1. If you were to zoom out on the fractal, it would appear to be a pattern of lines, therefore existing in one dimension. Thus, the self-similarity dimension, 1.46, is greater than the topological dimension, 1. This relationship (self-similiarity dimension > topological dimension) proves to be true for all fractals and is often included in the formal definition of these geometric shapes. The self-similarity dimension can be calculated with the method described above for mathematically created fractals that are exactly self-similar over different scales, however a different process called the Box-Counting Method [4] is used to calculate the dimension for more complex fractals, such as random (Section 3.1) or natural fractals, that are not exactly self-similar. Yet, the concept of dimension still holds true and is extremely important in defining all fractals. 3 More Complex Fractals And Applications With a background of fractals and how they are defined, we can look at more complex fractals that give an interesting insight into how these geometric shapes prove to be useful. 3.1 Random Fractals As we continue our exploration into fractals, it s important to consider how this geometric shape can be useful in a real-world application. Unfortunately, fractals that appear in the real world are usually not perfectly self-similar like the fractal snowflake we ve been considering. However, we can still utilize the simple iterative functions that create fractals to model natural phenomena by introducing the concept of randomness. Consider the Koch Curve, a classic iterative fractal in Figure 3 (a). With every iteration, the pattern is curved upwards. However, consider if we added randomness such that we curve upwards with a probability of 50% and downwards with a probability of 50%. This can be modeled by a simple statistical equation that produces a much more random shape such as in Figure 3 (b). The addition of randomness allows the curve to 3

model a more realistic situation that may be found in nature, such as the crack in a sidewalk or a coastline. (a) Classic Koch curve without randomness (b) Koch curve with randomness Figure 3: A comparison of the Koch curve with and without randomness [4] It s important to note that random fractals, like fractals found in nature, are not exactly self-similar. However, since the shape is made of smaller parts that have the same statistical properties as the whole, these shapes are considered statistically self-similar, and therefore still fractals by definition. The measurement of dimension is equally important in characterizing random fractals as it is for perfectly self-similar fractals, however, as previously mentioned, will need to be calculated with a slightly more advanced method than that introduced in the previous section. 3.2 Julia Sets and the Mandelbrot Set The Mandelbrot set, like random fractals, is a complex application of fractal ideas that proves to be an interesting area of study within this subject. Before introducing the Mandelbrot set, we must first define a Julia set. A Julia set is a set of conditions for a function f which, when iterated, do not go to infinity. For a simple example, consider the equation f(x) = x 2. When iterated, this function converges for initial conditions, x 0, of absolute value less than 1. Any x 0 > 1 will fly off to infinity when iterated. Thus, the Julia set for this function is [ 1, 1] since all numbers between 1 and 1, including the end points, will not go to infinity when iterated. When graphed on a complex plane, the Julia set for f(x) = x 2 is a simple circle with radius 1. The graph, however, will become more interesting when we consider more complicated equations of the form f(x) = x 2 + c where c is any complex number. In Section 2.1, we explored iterative functions and their use in defining fractals. Thus it shouldn t be surprising that in graphing the Julia sets of functions of the format f(x) = x 2 + c, we can produce complex fractals. In analyzing these fractals, it was discovered that there are two categories into which the graphs fall: connected or disconnected. It turns out that whether or not the graph is connected is dependent on if x 0 = 0 is included in the Julia set. If 0 is in the Julia set, then the graph will be connected, otherwise the fractal is disconnected. From this distinction arises another fractal: the Mandelbrot set. The Mandelbrot set is the collection of values of c such that f(x) = x 2 + c produces a connected set. Graphing the points of the Mandelbrot set on a complex plane, we find that 4

(a) The entire Mandelbrot set on the (b) An example of how the Mandelbrot set complex plane [6]. can be colored [7] Figure 4: Two images of the Mandelbrot set. this graph is also a fractal (Figure 4). Zooming in on various sections of the Mandelbrot set graph reveals infinitely complex patterns with interesting connections to the related Julia set [4]. The Mandelbrot set is often graphed with different colors (Figure 4 (b)) used to identify the rate at which certain values outside the set fly off to infinity. This creates more interesting and artistic images that demonstrate the complexity of the set, as well as look aesthetically pleasing to the viewer. Not only is the Mandelbrot set a beautiful mathematical set used in art and graphic design, but it provides a prime example of how simple rules can produce infinitely complex results. This is perhaps the most important reason for the study of fractals. These complex shapes, produced by simple rules, allow us to model complicated shapes that would seem otherwise too complex to reproduce without an understanding of fractals. 3.3 Applications of Fractals The applications of fractals can be seen in various fields of study from computer science to medicine. As mentioned, random fractals can be used to model coastlines, for example, which is useful for ecologists to mimic natural landscapes and thus form conclusions about natural habitat patterns [8]. Besides conducting important research that resulted in the discovery of the Mandelbrot set, Benoit Mandelbrot analyzed how computer produced random fractals can be used to represent patterns in nature such as mountain ranges [9]. Video games and movie animation alike produce fractal landscapes using the dimension of a random fractal. Fractals are also commonly used in image compression [10], geology and geophysics to model earthquakes, physiology and medicine in tissue imaging, and more. 4 Presentation Reflection and Comments For both this paper and my in class presentation, my main source of research was [4]. As this is a fairly large textbook, I pieced together the sections of information which I found most useful and hoped would be the most interesting to present. Overall, the comments from the class were favorable and I received very positive feedback on the material that was presented. Reflecting on my presentation, some classmates mentioned they would have liked further detail on the applications of fractals since I simply give an overview of many different applications. While I agree there are many interesting applications of fractals that would be fun to learn about in greater detail, my main motivation in giving this presentation was for the audience to understand fractals on a fundamental level and explore their creation and appearances in mathematics. I chose to center my research on the pure mathematics of this topic as I find the connections between iterative equations and fractal shape, as well as the phenomena of the fractals that appear when 5

graphing the Julia sets and Mandelbrot set, truly fascinating. Whereas the central mathematics I presented may not be about the direct applications of fractals, I think this deep understanding of fractals is necessary before one can dive into applying them to real world situations. Throughout this paper and my presentation, I nod to the applications of fractals and the adjustments that must be made to apply them to various situations. I hope this has sparked the interest of the audience to continue with their own research into the application that most speaks to them. Our class contains a variety of students who study math along with different topics such as psychology, computer science, economics, physics, etc. Some students noted they were interested in learning more about image compression, while others were more interested in the connection to fractal landscapes in computer imaging. Another student noted the connection to a TV show, Numb3ers, which comments on how fractals can be used to identify a forged comic book [11]. A separate presentation could dive deeply into any of these topics once the ground work of basic fractals is laid out. I encourage the audience of this paper or of my in-class presentation to choose a fractal application that most appeals to them and do some quick research. I have provided some sources in my reference list below but also note there are hundreds of sources on fractal applications that are just as interesting as the essays I have listed. In a course that holds the title mathematical connections, I think it s important to recognize the many connections a simple topic can have outside of pure mathematics even without going into extreme detail about each connection. This course has opened my eyes to the many discrete ways in which mathematics appears in our world. Even without a thorough knowledge of each of the mathematical connections, I appreciate the opportunity to hear about vastly different topics that spark my interest and desire to continue more detailed research on my own time. For me, learning the basic concepts of fractals reflected the overarching themes I have found throughout this course. Fractals are complex shapes and patterns that can be boiled down to a simple equation through elementary mathematics and I have found throughout this course that there are so many other things in life that can be similarly simplified with math. Thus, the study of fractals can truly force us to look at the world through a new mathematical lens that can apply to many other situations as we have analyzed throughout this course. References [1] Tree image. https://d2gg9evh47fn9z.cloudfront.net/800px COLOURBOX9099251.jpg [2] Romanesco Broccoli image. https://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/fractal Broccoli.jpg/220px- Fractal Broccoli.jpg [3] Fractal Foundation What Are Fractals?. https://fractalfoundation.org/resources/what-arefractals/ [4] David P. Feldman, Chaos and Fractals: An Elementary Introduction. Oxford University Press, 2012. http://web.a.ebscohost.com/ehost/ebookviewer/ebook?sid=762e720e- 73bf-4e48-b4ff-ea4a92fa8aef %40sessionmgr4010&vid=0&format=EB [5] Fractal Tree image. https://assets.wired.com/photos/w 851/wp-content/uploads/2014/12/ch08 04.png [6] Mandelbrot Set graph. https://www.cs.princeton.edu/ wayne/mandel/mandel-bw.jpg [7] Colored Mandelbrot Set graph. https://i.pinimg.com/originals/42/f6/e1/42f6e13b702ba3284cda63ed81d1fcd2.jpg [8] George Sugihara, Robert M. May, Applications of Fractals in Ecology. Trends in Ecology & Evolution 5 : 3, 1990, 79 86. https://www.sciencedirect.com/science/article/pii/0169534790902356 [9] Benoit B. Mandelbrot, Fractals and the Geometry of Nature. https://users.math.yale.edu/mandelbrot/web pdfs/encyclopediabritannica.pdf 6

[10] Lester Thomas, Farzin Deravi, Region-Based Fractal Image Compression Using Heuristic Search. IEE Transations on Image Processing 4 : 6, 1995, 832 838. https://ieeexplore.ieee.org/document/388086#full-text-section [11] Numb3rs 409: Graphic http://pi.math.cornell.edu/ numb3rs/baker/409.html 7