The Mandelbrot Fractal: An Imaginary JourneyLongphi NguyenKevin NelsonCollege of the RedwoodsDecember 20, 2010Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyTowards a Mandelbrot FractalFigure: A Mandelbrot FractalLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyThe Background of the Mandelbrot FractalFigure: Mandelbrot FractalOf all fractals, this is one of the mostfamous and well knownIt was discovered by BenoitMandelbrot in 1980It exists on the Argand planeAs a mathematical equation, it isgenerated by the recursion formula:zn+1= z2n+ cM = {c ∈ C| limn→∞|zn| 6= ∞}c = a + biLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyArgand PlaneReal axisImaginary axis(a, bi)Figure: Argand PlaneAlong the horizontal axis arethe real numbersAlong the vertical plane lay theimaginary numbersThe Argand plane is alsoreferred to as the complexplaneLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyComplex Number Magnitude: |zn|The m agnitude of a complexnumb er can also be called the”modulus”The m agnitude is computedusing:√a2+ b2RealImaginary(0, 0)ba(a, bi)Figure: MagnitudeLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyThe Iteration Formula: zn+1= z2n+ cInitializationc corresponds to some point (a, bi) on the Argand plane suchthat c = a + biz0is initialized with the beginning value of zeroFirst Iterationz1is assigned the value of z20+ c. Since z0= 0, z1= c|z1| is computedSecond Iterationz2is assigned the value of z21+ c|z2| is computedThird Iterationz3is assigned the value of z22+ c|z3| is computedLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyBounded and UnboundedEach of the following iterations in zn+1= z2n+ c are checked forbeing either bounded or unboundedBounded magnitudes will always be less than or equal to two, nomatter how many iterations are performed: limn→∞|zn| ≤ 2Unbounded magnitudes will go off to infinity, though they mayinitially be less than two: limn→∞|zn| > 2The following examples will illustrate how this happensLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyUnbounded Iteration Examplezn+1= z2n+ cLet c = 0.6 - 1.25iSet z0= 0First Iterationz1= (0)2+ (0.6 − 1.25i) = 0.6 − 1.25iSecond Iterationz2= (0.6 − 1.25i )2+ (0.6 − 1.25i) = −0.6025 − 2.75iThird Iterationz3= (−0.6025 − 2.75i )2+ (0.6 + 1.25i) = −6.5995 + 2.0638iForth Iterationz4= (−6.5995+2.0638i)2+(0.6+1.25i) = 39.8943−28.4894iLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyUnbounded: |zn| > 2At each step we check the magnituden zn|zn|0 0 01 0.6 − 1.25i 1.3872 −0.6025 − 2.7500i 2.8153 −6.5995 + 2.0638i 6.9154 39.8943 − 28.4894i 49.2Note how the magnitude is explodingLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyIllustration of Being UnboundedReal axisImaginary axis(1) (.6 + 1.25i )(2) (−.6025, 2.5)(3) (−6.60, −2.06)(4) (39.89, 28.94)Figure: Iteration TravelsLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyBounded: |zn| ≤ 2c = .2 + .3i: At each step, we check the magnituden zn|zn|0 0 01 .2+.3i .36062 .1500 + .4200i .44603 .0461 + .4260i .42854 0.0206 + .3391i .33995 0.853 + .3410i .32546 0.1087 + .35361i .3699.........11 .0851 + .3587i .3687.........49 .0792 + .3565i .365250 .0792 + .3565i .3652Note how znis approaching some value and the magnitude is alsoapproaching upon a single value.Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyFractal Programs and IterationsThere are three major ways to restrict iterations:Set a maximum number of iterationsMagnitude restrictionToleranceAll three of the above are usually user settable parametersOur MATLAB code uses a combination of maximumiterations and magnitude restriction to maximize speed forfractal creationLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyMaximum Iterations ExampleSet a maximum number of iterations to50Iterate up to 50 times for eachp ointIteration count stops:If |zk| > 2, stop andrecord iteration count (kvalue)If iterations equal 50, stopand record k = 50Recorded iteration countdetermines colorColor is determined by the chosencolor schemeThe recorded iteration count is alsoreferred to as ”depth”Figure: An Unbounded IterationOutside and inside red: How different?Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyPortion of Mandelbrot ExaminedOur future examples will be within the left stem The examples willuse a 5 x 5 matrixFigure: A Mandelbrot Fractal Figure: Left StemLongphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyMatrix of Initial Zerosz0=0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0A common size is a 500 x 500 matrix, 250,000 discrete points.Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyFirst Iteration: z1= cz1= c−1.90 + 0.20i −1.80 + 0.20i −1.70 + 0.20i −1.60 + 0.20i −1.50 + 0.20i−1.90 + 0.10i −1.80 + 0.10i −1.70 + 0.10i −1.60 + 0.10i −1.50 + 0.10i−1.90 −1.80 −1.70 −1.60 −1.50−1.90 − 0.10i −1.80 − 0.10i −1.70 − 0.10i −1.60 − 0.10i −1.50 − 0.10i−1.90 − 0.20i −1.80 − 0.20i −1.70 − 0.20i −1.60 − 0.20i −1.50 − 0.20iMagnitude : |z1| =1.91 1.81 1.71 1.61 1.511.90 1.80 1.70 1.60 1.501.9 1.8 1.7 1.6 1.51.90 1.80 1.70 1.60 1.501.91 1.81 1.71 1.61 1.51Iteration Count =1 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 1Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneySecond Iteration: z2z2=1.67 − 0.56i 1.40 − 0.52i 1.15 − 0.48i 0.92 − 0.44i 0.71 − 0.40i1.70 − 0.28i 1.43 − 0.26i 1.18 − 0.24i 0.95 − 0.22i 0.76 − 0.20i1.71 1.44 1.19 0.96 0.751.70 + 0.28i 1.43 + 0.26i 1.18 + 0.24i 0.95 + 0.22i 0.76 + 0.20i1.67 + 0.56i 1.40 + 0.52i 1.15 + 0.48i 0.92 + 0.44i 0.71 + 0.40iMagnitude : |z2| =1.7614 1.4935 1.2462 1.0198 0.814921.7229 1.4534 1.2042 0.97514 0.766551.71 1.44 1.19 0.96 0.751.7229 1.4534 1.2042 0.97514 0.766551.761 1.4935 1.2462 1.0198 0.81492Iteration Count =2 2 2 2 22 2 2 2 22 2 2 2 22 2 2 2 22 2 2 2 2Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary JourneyThird Iteration: z3z3=0.57 − 1.67i −0.11 − 1.25i −0.60 − 0.90i −0.94 − 0.60i −1.15