BackgroundVisualizing FractalsMagnitudeBounded Magnitude Results Through IterationMandelbrot SetBounded and Unbounded Magnitudes for the Mandelbrot SetMATLAB Function for Mandelbrot FractalParametersInitialized ValuesCalculationsColoringBackgroundVisualizing FractalsMandelbrot SetMATLAB Function for . . .Home PageTitle PageJJ IIJ IPage 1 of 22Go BackFull ScreenCloseQuitFractal Visualizations on the Complex PlaneLongphi Nguyen and Kevin NelsonDecember 20, 2010AbstractThis paper will detail the visualization of fractals by using the MandelbrotFractal as the main focus. MATLAB is used to produce the visualizations of theMandelbrot Fractal. The code function, detailed within the paper, uses matrices asa basis for graphing the fractals. Subjects discussed in the paper include boundedand unbounded magnitudes, the Argand plane, iteration, and the MandelbrotFractal.BackgroundIsaac Newton developed the first fractal equation using a method now referred to asNewton’s Method. But lacking computing power, he did not know the full extent ofwhat he had developed.The term “fractal” was given by Benoit Mandelbrot in 1975. The term fits welldue to the fragmented geometric shapes that fractals usually have. Typically, frac-tals are formed from rather simple functions that undergo many – easily millions – ofcalculations through a process known as iteration.BackgroundVisualizing FractalsMandelbrot SetMATLAB Function for . . .Home PageTitle PageJJ IIJ IPage 2 of 22Go BackFull ScreenCloseQuitVisualizing FractalsMagnitudeThe magnitude of real numbers from the origin is simple. It is the square root ofthe sum of the squares of the components. For for a point (a, b) where a, b, ∈ R, themagnitude is√a2+ b2. But the magnitude of complex numbers is slightly different thanthe real numbers case. For complex numbers, there is a real part and an imaginarypart. For example, c = a + bi has the real part a and the imaginary part bi. However,the magnitude of c is notpa2+ (bi)2. Rather, we leave out the i. The magnitude of cis |c| =√a2+ b2.To plot complex numbers, we use the Argand Plane. The Argand Plane has a realaxis and an imaginary axis. The real part a of a complex number determines whereto plot a point with respect to the real axis. Similarly, the coefficient in the imaginarypart, b, determines where to plot a point with respect to the imaginary axis. Figure 1shows how the Pythagorean Theorem gives the distance formula for complex numbersin the Argand plane.Bounded Magnitude Results Through IterationFractals have become a new sensation due to their fascinating images. As stated earlier,fractal images are created through many calculations. The following explanation usesa simple case. Suppose a function f(z) is given. For some value of a constant z0, f(z0)returns a value z1. Then do the calculation again using z1such that f(z1) = z2.Repeat the process so that f(zn) = zn+1. The process of repeating calculation isknown as an iteration process. Iteration is essential for graphing fractals. When aninitial z0eventually leads to outputs with magnitudes that approach infinity, the z0issaid to lead to unbounded results. Similarly, when an initial z0value does not lead toan unbounded result, it is referred to as a bounded result. Bounded and unboundedBackgroundVisualizing FractalsMandelbrot SetMATLAB Function for . . .Home PageTitle PageJJ IIJ IPage 3 of 22Go BackFull ScreenCloseQuitRealImaginary(0, 0)ba(a, bi)Figure 1: Magnitude√a2+ b2magnitudes will be of great importance, so let us define them again:unbounded : limn→∞|zn| = ∞bounded : limn→∞|zn| 6= ∞As an example, suppose z0∈ C. Note that complex z0values are typical whenworking with fractals. Let z0= 1.1 + 0.74i and f(zn) = zn+1= z2n. In the Argandplane, this z0is associated with 1.1 on the real axis and 0.74i on the imaginary axis(refer to Figure 2).Now iterate z0through the function f(z) several times as to determine if the z0leads to results that have bounded magnitude.BackgroundVisualizing FractalsMandelbrot SetMATLAB Function for . . .Home PageTitle PageJJ IIJ IPage 4 of 22Go BackFull ScreenCloseQuitReal axisImaginary axis(1.1, 0.74i)Figure 2: Argand Planez0 = 1.1 + .74i; magnitude = 1.3257z1 = z02= .6624 + 1.6280i; magnitude = 1.7576z2 = z12= −2.2116 + 2.1568i; magnitude = 3.0892z3 = z22= .2395 − 9.5399i; magnitude = 9.5429The magnitude increases as we iterate. Thus, the initial z0leads to results withunbounded magnitude. As another example, let z0= 0.40+0.1i and f(zn) = zn+1= z2n.Iterating through the function gives the outputsBackgroundVisualizing FractalsMandelbrot SetMATLAB Function for . . .Home PageTitle PageJJ IIJ IPage 5 of 22Go BackFull ScreenCloseQuitz0 = .4 + .1i; magnitude = .4123z1 = .15 + .08i; magnitude = .1700z2 = .0161 + .024i; magnitude = .0289It appears that the initial z0is leading to results with bounded magnitude.Mandelbrot SetThe Mandelbrot set, denoted by M, has a rather simple definition, but an image of theMandelbrot Set can be quite astonishing. The definition is as follows:M = {c ∈ C| limn→∞|zn| 6= ∞}andzn+1= z2n+ cwherez0= 0,c = a + bi.Notice in the definition that magnitude is used to determine if a c value leads toresults that are in the Mandelbrot set.BackgroundVisualizing FractalsMandelbrot SetMATLAB Function for . . .Home PageTitle PageJJ IIJ IPage 6 of 22Go BackFull ScreenCloseQuitFigure 3: Visualization of the Mandelbrot SetBounded and Unbounded Magnitudes for the Mandelbrot SetFor the Mandelbrot set, we often use magnitude as a means for determining if resultsare bounded or unbounded. If the iteration process continues to produce outputs withlarge magnitudes for many iterations, it may be showing an unbounded result. If theBackgroundVisualizing FractalsMandelbrot SetMATLAB Function for . . .Home PageTitle PageJJ IIJ IPage 7 of 22Go BackFull ScreenCloseQuitReal axisImaginary axis(.6, 1.25)Figure 4: A Specific Pointmagnitudes fluctuate back and forth throughout many iterations or closes in to a singlevalue, further iterations may show a bounded result. However, whether the iterationprocess leads to bounded results or to unbounded results is not always obvious after afew iterations; sometimes it may require hundreds or more iterations to know precisely.When using the magnitude of complex numbers, determining if results are boundedor unbounded is unpredictable. The iteration of complex numbers is a type of dynamicalsystem; we are taking an initial complex number and repeatedly