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

Figure: 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: zn+1 = z2

n + c

M = {c ∈ C| limn→∞ |zn| = ∞} c = a + bi

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Argand Plane

Real axis Imaginary axis (a, bi)

Figure: Argand Plane Along the horizontal axis are the real numbers Along the vertical plane lay the imaginary numbers The Argand plane is also referred to as the complex plane

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Complex Number Magnitude: |zn|

The magnitude of a complex number can also be called the ”modulus” The magnitude is computed using: √ a2 + b2

Real Imaginary (0, 0) b a (a, bi)

Figure: Magnitude

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

The Iteration Formula: zn+1 = z2

n + c Initialization c corresponds to some point (a, bi) on the Argand plane such that c = a + bi z0 is initialized with the beginning value of zero First Iteration z1 is assigned the value of z2

0 + c. Since z0 = 0, z1 = c

|z1| is computed Second Iteration z2 is assigned the value of z2

1 + c

|z2| is computed Third Iteration z3 is assigned the value of z2

2 + c

|z3| is computed

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Bounded and Unbounded

Each of the following iterations in zn+1 = z2

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: limn→∞ |zn| ≤ 2 Unbounded magnitudes will go off to infinity, though they may initially be less than two: limn→∞ |zn| > 2 The following examples will illustrate how this happens

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Unbounded Iteration Example

zn+1 = z2

n + c

Let c = 0.6 - 1.25i Set z0 = 0 First Iteration

z1 = (0)2 + (0.6 − 1.25i) = 0.6 − 1.25i

Second Iteration

z2 = (0.6 − 1.25i)2 + (0.6 − 1.25i) = −0.6025 − 2.75i

Third Iteration

z3 = (−0.6025 − 2.75i)2 + (0.6 + 1.25i) = −6.5995 + 2.0638i

Forth Iteration

z4 = (−6.5995+2.0638i)2+(0.6+1.25i) = 39.8943−28.4894i

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Unbounded: |zn| > 2

At each step we check the magnitude n zn |zn| 1 0.6 − 1.25i 1.387 2 −0.6025 − 2.7500i 2.815 3 −6.5995 + 2.0638i 6.915 4 39.8943 − 28.4894i 49.2 Note how the magnitude is exploding

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Illustration of Being Unbounded

Real axis Imaginary axis

(1) (.6 + 1.25i) (2) (−.6025, 2.5) (3) (−6.60, −2.06) (4) (39.89, 28.94)

Figure: Iteration Travels

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Bounded: |zn| ≤ 2

c = .2 + .3i: At each step, we check the magnitude

n zn |zn| 1 .2+.3i .3606 2 .1500 + .4200i .4460 3 .0461 + .4260i .4285 4 0.0206 + .3391i .3399 5 0.853 + .3410i .3254 6 0.1087 + .35361i .3699 . . . . . . . . . 11 .0851 + .3587i .3687 . . . . . . . . . 49 .0792 + .3565i .3652 50 .0792 + .3565i .3652

Note how zn 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 |zk| > 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 The recorded iteration count is also referred to as ”depth”

Figure: An Unbounded Iteration

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

z0 =             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: z1 = c

z1 = 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 1.51 1.90 1.80 1.70 1.60 1.50 1.9 1.8 1.7 1.6 1.5 1.90 1.80 1.70 1.60 1.50 1.91 1.81 1.71 1.61 1.51

    Iteration Count =    

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

   

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Second Iteration: z2

z2 =   

1.67 − 0.56i 1.40 − 0.52i 1.15 − 0.48i 0.92 − 0.44i 0.71 − 0.40i 1.70 − 0.28i 1.43 − 0.26i 1.18 − 0.24i 0.95 − 0.22i 0.76 − 0.20i 1.71 1.44 1.19 0.96 0.75 1.70 + 0.28i 1.43 + 0.26i 1.18 + 0.24i 0.95 + 0.22i 0.76 + 0.20i 1.67 + 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 0.81492 1.7229 1.4534 1.2042 0.97514 0.76655 1.71 1.44 1.19 0.96 0.75 1.7229 1.4534 1.2042 0.97514 0.76655 1.761 1.4935 1.2462 1.0198 0.81492

    Iteration Count =    

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

   

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Third Iteration: z3

z3 =   

0.57 − 1.67i −0.11 − 1.25i −0.60 − 0.90i −0.94 − 0.60i −1.15 − 0.36i 0.91 − 0.85i 0.17 − 0.64i −0.36 − 0.46i −0.74 − 0.31i −0.97 − 0.20i 1.0241 0.2736 −0.2839 −0.6784 −0.9375 0.91 + 0.85i 0.17 + 0.64i 0.36 + 0.46i 0.74 + 0.31i 0.97 + 0.20i 0.57 + 1.67i 0.11 + 1.25i 0.60 + 0.90i 0.94 + 0.60i 1.15 + 0.36i

  

Magnitude : |z3| =    

1.7667 1.2608 1.0894 1.1264 1.2131 1.2478 0.66757 0.59237 0.81086 1.0116 1.0241 0.2736 0.2839 0.6784 0.9375 1.2478 0.66757 0.59237 0.81086 1.0116 1.7667 1.2608 1.0894 1.1264 1.2131

    Iteration Count =    

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

   

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Fourth Iteration: z4

z4 =   

−4.35 − 1.72i −3.36 + 0.47i −2.14 + 1.29i −1.07 + 1.35i −0.29 + 1.05i −1.79 − 1.45i −2.18 − 0.12i −1.78 + 0.44i −1.14 + 0.57i −0.60 + 0.50i −0.8512 −1.7251 −1.6194 −1.1398 −0.6211 −1.79 + 1.45i −2.18 + 0.12i −1.78 − 0.44i −1.14 − 0.57i −0.60 − 0.50i −4.35 + 1.72i −3.36 − 0.47i −2.14 − 1.29i −1.07 − 1.35i −0.29 − 1.05i

  

Magnitude : |z4| =    

4.687 3.399 2.51 1.7291 1.0925 2.3095 2.1865 1.8378 1.2808 0.73863 0.85122 1.7251 1.6194 1.1398 0.62109 2.3095 2.1865 1.8378 1.2808 0.73863 4.687 3.399 2.51 1.7291 1.0925

    Iteration Count =    

3 3 3 4 4 3 3 4 4 4 4 4 4 4 4 3 3 4 4 4 3 3 3 4 4

   

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Fifth Iteration: z5

z5 =   

14.13 + 15.21i 9.29 − 3.01i 1.22 − 5.38i −2.28 − 2.71i −2.51 − 0.42i −0.79 + 5.31i 2.94 + 0.65i 1.28 − 1.47i −0.61 − 1.21i −1.39 − 0.50i −1.1754 1.1761 0.9225 −0.3009 −1.1142 −0.79 − 5.31i 2.94 − 0.65i 1.28 + 1.47i −0.61 + 1.21i −1.39 + 0.50i 14.13 − 15.21i 9.29 + 3.01i 1.22 + 5.38i −2.28 + 2.71i −2.51 + 0.42i

  

Magnitude : |z5| =    

20.7682 9.7737 5.5177 3.5433 2.5508 5.3757 3.0210 1.9569 1.3639 1.4859 1.1754 1.1761 0.9225 0.3009 1.1142 5.3757 3.0210 1.9569 1.3639 1.4992 20.7682 9.7737 5.5177 3.5433 2.5508

    Iteration Count =    

3 3 3 4 4 3 3 5 5 5 5 5 5 5 5 3 3 5 5 5 3 3 3 4 4

   

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Second and Fiftieth Iteration Comparisons: z2 and z50

z2 =   

1.67 − 0.56i 1.40 − 0.52i 1.15 − 0.48i 0.92 − 0.44i 0.71 − 0.40i 1.70 − 0.28i 1.43 − 0.26i 1.18 − 0.24i 0.95 − 0.22i 0.76 − 0.20i 1.71 1.44 1.19 0.96 0.75 1.70 + 0.28i 1.43 + 0.26i 1.18 + 0.24i 0.95 + 0.22i 0.76 + 0.20i 1.67 + 0.56i 1.40 + 0.52i 1.15 + 0.48i 0.92 + 0.44i 0.71 + 0.40i

  

z50 = NaN = Not a Number Just too big!    

NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi −1.5785 1.4153 0.45479 −1.4166 0.61531 NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi

   

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Iteration Chaos

The bounded values are often different for each iteration. The following values are from the real number axis of the previous examples. Note how they are all bounded yet unsettled

z1 : −1.90 −1.80 −1.70 −1.60 −1.50 z101 : 0.26748 0.7738 −1.6092 −0.36454 −0.24833 z102 : −1.8285 −1.2012 0.88944 −1.4671 −1.4383 z103 : 1.4432 −0.35703 −0.90889 0.55241 0.56879 z104 : 0.18297 −1.6725 −0.87392 −1.2948 −1.1765 z105 : −1.8665 0.99735 −0.93627 0.076621 −0.11591 z106 : 1.5839 −0.8053 −0.82339 −1.5941 −1.4866 z107 : 0.60876 −1.1515 −1.022 0.94125 0.70987

IterationCount at z6

   

3 3 3 4 4 3 3 5 5 6 6 6 6 6 6 3 3 5 5 6 3 3 3 4 4

   

IterationCount at z107   

3 3 3 4 4 3 3 5 5 6 107 107 107 107 107 3 3 5 5 6 3 3 3 4 4

  

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Magnitude Contrasts at z5 and z7

Magnitude at z5 =   

20.7682 9.7737 5.5177 3.5433 2.5508 5.3757 3.0210 1.9569 1.3639 1.4859 1.1754 1.1761 0.9225 0.3009 1.1142 5.3757 3.0210 1.9569 1.3639 1.4992 20.7682 9.7737 5.5177 3.5433 2.5508

   Magnitude at z7 =   

1.8633e + 005 8826.5 1016.9 173.28 26.384 940.55 56.91 19.477 9.0814 3.4092 1.6313 1.626 0.9790 0.6784 1.4332 940.55 56.91 19.477 9.0814 3.4092 1.8633e + 005 8826.5 1016.9 173.28 26.384

  

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

The Mandelbrot When Viewed Sideways

Figure: Mandelbrot Fractal Side Image

Here we see the Mandelbrot from a different perspective. The structure of the fringes is clearly seen The top areas are where the values are bounded The lower areas are where the values are unbounded

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

A Mandelbrot Spiral

Figure: The Spinners

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Mandelbrot Snowflakes

Figure: Stuck Snowflakes

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

The Mandelbrot Within the Mandelbrot

Figure: ZaoS Loc (-.0582788824913,-0.443337214170), Size 3.2e-5

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

The −π Multibrot Fractal

Figure: Mandelbrot Fractal zn+1 = z−π

n

+ c All real numbers can be an exponent in the equation zn+1 = zx

n + c,

to create a “Multibrot,” x = 2. Only zn+1 = z2

n + c is

a Mandelbrot

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Common Alternate Multibrots

Figure: zn+1 = z4

n + c

Figure: zn+1 = z7

n + c

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey

Bibliography

Arnold, Dave. Writing Scientific Papers in L

AT

EX The BEAMER class User Guide for version 3.10 Darling, David The Universal Book of Mathematics, Wiley, 2004 Devaney,Robert L., Keen, Linda ed. Chaos and Fractals: The Mathematics Behind the Computer Graphics Proceedings of Symposia in Applied Mathematics, American Mathmatical Society, Volume 39, 1980 Gr¨ atzer, George. More Math Into L

AT

EX, 4th ed. Springer, 2007 Lamport, Leslie. L

AT

EX A Document Preparation System, 2nd

• ed. Addison Wesley, 1985

Longphi Nguyen Kevin Nelson The Mandelbrot Fractal: An Imaginary Journey