The Mandelbrot Fractal: An Imaginary Journey Longphi Nguyen Kevin - PowerPoint PPT Presentation

The Mandelbrot Fractal: An Imaginary Journey Longphi Nguyen Kevin Nelson College of the Redwoods December 20, 2010 Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Towards a Mandelbrot Fractal Figure: A Mandelbrot Fractal Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

The Background of the Mandelbrot Fractal Of all fractals, this is one of the most famous and well known It was discovered by Benoit Mandelbrot in 1980 It exists on the Argand plane As a mathematical equation, it is generated by the recursion formula: z n +1 = z 2 n + c M = { c ∈ C | lim n →∞ | z n | � = ∞} c = a + bi Figure: Mandelbrot Fractal Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Argand Plane Imaginary axis ( a , bi ) Along the horizontal axis are the real numbers Real axis Along the vertical plane lay the imaginary numbers The Argand plane is also referred to as the complex plane Figure: Argand Plane Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Complex Number Magnitude: | z n | Imaginary ( a , bi ) b The magnitude of a complex number can also be called the ”modulus” Real a (0 , 0) The magnitude is computed √ a 2 + b 2 using: Figure: Magnitude Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

The Iteration Formula: z n +1 = z 2 n + c Initialization c corresponds to some point ( a , bi ) on the Argand plane such that c = a + bi z 0 is initialized with the beginning value of zero First Iteration z 1 is assigned the value of z 2 0 + c . Since z 0 = 0 , z 1 = c | z 1 | is computed Second Iteration z 2 is assigned the value of z 2 1 + c | z 2 | is computed Third Iteration z 3 is assigned the value of z 2 2 + c | z 3 | is computed Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Bounded and Unbounded Each of the following iterations in z n +1 = z 2 n + c are checked for being either bounded or unbounded Bounded magnitudes will always be less than or equal to two, no matter how many iterations are performed: lim n →∞ | z n | ≤ 2 Unbounded magnitudes will go off to infinity, though they may initially be less than two: lim n →∞ | z n | > 2 The following examples will illustrate how this happens Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Unbounded Iteration Example z n +1 = z 2 n + c Let c = 0.6 - 1.25i Set z 0 = 0 First Iteration z 1 = (0) 2 + (0 . 6 − 1 . 25 i ) = 0 . 6 − 1 . 25 i Second Iteration z 2 = (0 . 6 − 1 . 25 i ) 2 + (0 . 6 − 1 . 25 i ) = − 0 . 6025 − 2 . 75 i Third Iteration z 3 = ( − 0 . 6025 − 2 . 75 i ) 2 + (0 . 6 + 1 . 25 i ) = − 6 . 5995 + 2 . 0638 i Forth Iteration z 4 = ( − 6 . 5995+2 . 0638 i ) 2 +(0 . 6+1 . 25 i ) = 39 . 8943 − 28 . 4894 i Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Unbounded: | z n | > 2 At each step we check the magnitude | z n | n z n 0 0 0 1 0 . 6 − 1 . 25 i 1.387 2 − 0 . 6025 − 2 . 7500 i 2.815 3 − 6 . 5995 + 2 . 0638 i 6.915 4 39 . 8943 − 28 . 4894 i 49.2 Note how the magnitude is exploding Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Illustration of Being Unbounded Imaginary axis (4) (39 . 89 , 28 . 94) (2) ( − . 6025 , 2 . 5) (1) ( . 6 + 1 . 25 i ) Real axis (3) ( − 6 . 60 , − 2 . 06) Figure: Iteration Travels Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Bounded: | z n | ≤ 2 c = . 2 + . 3 i : At each step, we check the magnitude | z n | n z n 0 0 0 1 .2+.3i .3606 2 . 1500 + . 4200 i .4460 3 . 0461 + . 4260 i .4285 4 0 . 0206 + . 3391 i .3399 5 0 . 853 + . 3410 i .3254 6 0 . 1087 + . 35361 i .3699 . . . . . . . . . 11 . 0851 + . 3587 i .3687 . . . . . . . . . 49 . 0792 + . 3565 i .3652 50 . 0792 + . 3565 i .3652 Note how z n is approaching some value and the magnitude is also approaching upon a single value. Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Fractal Programs and Iterations There are three major ways to restrict iterations: Set a maximum number of iterations Magnitude restriction Tolerance All three of the above are usually user settable parameters Our MATLAB code uses a combination of maximum iterations and magnitude restriction to maximize speed for fractal creation Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Maximum Iterations Example Set a maximum number of iterations to 50 Iterate up to 50 times for each point Iteration count stops: If | z k | > 2, stop and record iteration count ( k value) If iterations equal 50, stop and record k = 50 Recorded iteration count determines color Color is determined by the chosen color scheme Figure: An Unbounded Iteration The recorded iteration count is also referred to as ”depth” Outside and inside red: How different? Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Portion of Mandelbrot Examined Our future examples will be within the left stem The examples will use a 5 x 5 matrix Figure: A Mandelbrot Fractal Figure: Left Stem Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Matrix of Initial Zeros z 0 =  0 0 0 0 0  0 0 0 0 0     0 0 0 0 0     0 0 0 0 0   0 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 Journey

First Iteration: z 1 = c z 1 = c  − 1 . 90 + 0 . 20 i − 1 . 80 + 0 . 20 i − 1 . 70 + 0 . 20 i − 1 . 60 + 0 . 20 i − 1 . 50 + 0 . 20 i  − 1 . 90 + 0 . 10 i − 1 . 80 + 0 . 10 i − 1 . 70 + 0 . 10 i − 1 . 60 + 0 . 10 i − 1 . 50 + 0 . 10 i  − 1 . 90 − 1 . 80 − 1 . 70 − 1 . 60 − 1 . 50    − 1 . 90 − 0 . 10 i − 1 . 80 − 0 . 10 i − 1 . 70 − 0 . 10 i − 1 . 60 − 0 . 10 i − 1 . 50 − 0 . 10 i − 1 . 90 − 0 . 20 i − 1 . 80 − 0 . 20 i − 1 . 70 − 0 . 20 i − 1 . 60 − 0 . 20 i − 1 . 50 − 0 . 20 i Magnitude : | z 1 | = Iteration Count =     1 1 1 1 1 1 . 91 1 . 81 1 . 71 1 . 61 1 . 51 1 1 1 1 1 1 . 90 1 . 80 1 . 70 1 . 60 1 . 50     1 1 1 1 1 1 . 9 1 . 8 1 . 7 1 . 6 1 . 5     1 1 1 1 1    1 . 90 1 . 80 1 . 70 1 . 60 1 . 50  1 1 1 1 1 1 . 91 1 . 81 1 . 71 1 . 61 1 . 51 Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Second Iteration: z 2 z 2 =  1 . 67 − 0 . 56 i 1 . 40 − 0 . 52 i 1 . 15 − 0 . 48 i 0 . 92 − 0 . 44 i 0 . 71 − 0 . 40 i  1 . 70 − 0 . 28 i 1 . 43 − 0 . 26 i 1 . 18 − 0 . 24 i 0 . 95 − 0 . 22 i 0 . 76 − 0 . 20 i  1 . 71 1 . 44 1 . 19 0 . 96 0 . 75    1 . 70 + 0 . 28 i 1 . 43 + 0 . 26 i 1 . 18 + 0 . 24 i 0 . 95 + 0 . 22 i 0 . 76 + 0 . 20 i 1 . 67 + 0 . 56 i 1 . 40 + 0 . 52 i 1 . 15 + 0 . 48 i 0 . 92 + 0 . 44 i 0 . 71 + 0 . 40 i Magnitude : | z 2 | = Iteration Count =     2 2 2 2 2 1 . 7614 1 . 4935 1 . 2462 1 . 0198 0 . 81492 2 2 2 2 2 1 . 7229 1 . 4534 1 . 2042 0 . 97514 0 . 76655     2 2 2 2 2 1 . 71 1 . 44 1 . 19 0 . 96 0 . 75     2 2 2 2 2 1 . 7229 1 . 4534 1 . 2042 0 . 97514 0 . 76655     2 2 2 2 2 1 . 761 1 . 4935 1 . 2462 1 . 0198 0 . 81492 Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Third Iteration: z 3 z 3 =  0 . 57 − 1 . 67 i − 0 . 11 − 1 . 25 i − 0 . 60 − 0 . 90 i − 0 . 94 − 0 . 60 i − 1 . 15 − 0 . 36 i  0 . 91 − 0 . 85 i 0 . 17 − 0 . 64 i − 0 . 36 − 0 . 46 i − 0 . 74 − 0 . 31 i − 0 . 97 − 0 . 20 i  1 . 0241 0 . 2736 − 0 . 2839 − 0 . 6784 − 0 . 9375    0 . 91 + 0 . 85 i 0 . 17 + 0 . 64 i 0 . 36 + 0 . 46 i 0 . 74 + 0 . 31 i 0 . 97 + 0 . 20 i 0 . 57 + 1 . 67 i 0 . 11 + 1 . 25 i 0 . 60 + 0 . 90 i 0 . 94 + 0 . 60 i 1 . 15 + 0 . 36 i Magnitude : | z 3 | = Iteration Count =   3 3 3 3 3  1 . 7667 1 . 2608 1 . 0894 1 . 1264 1 . 2131  3 3 3 3 3 1 . 2478 0 . 66757 0 . 59237 0 . 81086 1 . 0116     3 3 3 3 3 1 . 0241 0 . 2736 0 . 2839 0 . 6784 0 . 9375     3 3 3 3 3    1 . 2478 0 . 66757 0 . 59237 0 . 81086 1 . 0116  3 3 3 3 3 1 . 7667 1 . 2608 1 . 0894 1 . 1264 1 . 2131 Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Fourth Iteration: z 4 z 4 =  − 4 . 35 − 1 . 72 i − 3 . 36 + 0 . 47 i − 2 . 14 + 1 . 29 i − 1 . 07 + 1 . 35 i − 0 . 29 + 1 . 05 i  − 1 . 79 − 1 . 45 i − 2 . 18 − 0 . 12 i − 1 . 78 + 0 . 44 i − 1 . 14 + 0 . 57 i − 0 . 60 + 0 . 50 i  − 0 . 8512 − 1 . 7251 − 1 . 6194 − 1 . 1398 − 0 . 6211    − 1 . 79 + 1 . 45 i − 2 . 18 + 0 . 12 i − 1 . 78 − 0 . 44 i − 1 . 14 − 0 . 57 i − 0 . 60 − 0 . 50 i − 4 . 35 + 1 . 72 i − 3 . 36 − 0 . 47 i − 2 . 14 − 1 . 29 i − 1 . 07 − 1 . 35 i − 0 . 29 − 1 . 05 i Magnitude : | z 4 | = Iteration Count =     3 3 3 4 4 4 . 687 3 . 399 2 . 51 1 . 7291 1 . 0925 3 3 4 4 4 2 . 3095 2 . 1865 1 . 8378 1 . 2808 0 . 73863     4 4 4 4 4 0 . 85122 1 . 7251 1 . 6194 1 . 1398 0 . 62109     3 3 4 4 4    2 . 3095 2 . 1865 1 . 8378 1 . 2808 0 . 73863  3 3 3 4 4 4 . 687 3 . 399 2 . 51 1 . 7291 1 . 0925 Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey