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 Fractal………………………………………………………………..……………………….6 Section IV Elementary Fractals………………………………………………………………..………………………….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 Set………………………………………………………………..………………………..22 a) Connectedness of the Mandelbrot Set………………………………………………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 mathematics; fractal geometry. 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, topology, 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 patterns 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 Computer 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% theorems 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 dimension strictly exceeds the topological dimension.” Topological dimension, DT (aka Lebesgue Dimension) is the minimum number of integers it takes to describe a shape. So, for a point, the topological dimension would be 0, for a line: 1, for a plane: 2, for a cube: 3 (height, length, width). Hausdorff dimension (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 iteration)/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 “fractal dimension” 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 complexity 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 Koch snowflake 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 iterations, 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 space. 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 function without a derivative.”8 This feature makes the von Koch arc a very distasteful curve to conventional