the mandelbrot fractal: an imaginary journey · 2010-12-20 · figure:mandelbrot fractal of all...

$: The Mandelbrot Fractal: An Imaginary Journey · 2010-12-20 · Figure:Mandelbrot Fractal Of all fractals, this is one of the most famous and well known It was discovered by Benoit$
The Mandelbrot Fractal: An Imaginary Journey

Longphi NguyenKevin 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 FractalLongphi 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 mostfamous and well known

It was discovered by BenoitMandelbrot in 1980

It exists on the Argand plane

As a mathematical equation, it isgenerated by the recursion formula:zn+1 = z2

n + c

M = {c ∈ C| limn→∞ |zn| 6=∞}

c = a + bi


Argand Plane

Real axis

Imaginary axis

(a, bi)

Figure: Argand Plane

Along the horizontal axis arethe real numbers

Along the vertical plane lay theimaginary numbers

The Argand plane is alsoreferred to as the complexplane


Complex Number Magnitude: |zn|

The magnitude of a complexnumber can also be called the”modulus”

The magnitude is computedusing:

√a2 + b2

Real

Imaginary

(0, 0)

b

a

(a, bi)

Figure: Magnitude


The Iteration Formula: zn+1 = z2n + c

Initialization

c corresponds to some point (a, bi) on the Argand plane suchthat c = a + biz0 is initialized with the beginning value of zero

First Iteration

z1 is assigned the value of z20 + c . Since z0 = 0, z1 = c

|z1| is computed

Second Iteration

z2 is assigned the value of z21 + c

|z2| is computed

Third Iteration

z3 is assigned the value of z22 + c

|z3| is computed


Bounded and Unbounded

Each of the following iterations in zn+1 = z2n + c are checked for

being either bounded or unbounded

Bounded magnitudes will always be less than or equal to two, nomatter how many iterations are performed: limn→∞ |zn| ≤ 2

Unbounded magnitudes will go off to infinity, though they mayinitially be less than two: limn→∞ |zn| > 2

The following examples will illustrate how this happens


Unbounded Iteration Example

zn+1 = z2n + c

Let c = 0.6 - 1.25iSet 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


Unbounded: |zn| > 2

At each step we check the magnitude

n 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 49.2

Note how the magnitude is exploding


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


Bounded: |zn| ≤ 2

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

n 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 .3652

Note how zn is approaching some value and the magnitude is alsoapproaching upon a single value.


Fractal Programs and Iterations

There are three major ways to restrict iterations:

Set a maximum number of iterationsMagnitude restrictionToleranceAll three of the above are usually user settable parameters

Our MATLAB code uses a combination of maximumiterations and magnitude restriction to maximize speed forfractal creation


Maximum Iterations Example

Set a maximum number of iterations to50

Iterate up to 50 times for eachpoint

Iteration count stops:

If |zk | > 2, stop andrecord iteration count (kvalue)If iterations equal 50, stopand record k = 50

Recorded iteration countdetermines color

Color is determined by the chosencolor scheme

The recorded iteration count is alsoreferred to as ”depth”

Figure: An Unbounded Iteration

Outside and inside red: How different?


Portion of Mandelbrot Examined

Our future examples will be within the left stem The examples willuse a 5 x 5 matrix

Figure: A Mandelbrot Fractal Figure: Left Stem


Matrix of Initial Zeros

z0 =

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.


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 1.511.90 1.80 1.70 1.60 1.501.9 1.8 1.7 1.6 1.5

1.90 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


Second Iteration: z2

z2 =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− 0.20i

1.71 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 0.76655

1.71 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


Third Iteration: z3

z3 =0.57− 1.67i −0.11− 1.25i −0.60− 0.90i −0.94− 0.60i −1.15− 0.36i0.91− 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 −0.93750.91 + 0.85i 0.17 + 0.64i 0.36 + 0.46i 0.74 + 0.31i 0.97 + 0.20i0.57 + 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 1.21311.2478 0.66757 0.59237 0.81086 1.01161.0241 0.2736 0.2839 0.6784 0.93751.2478 0.66757 0.59237 0.81086 1.01161.7667 1.2608 1.0894 1.1264 1.2131

Iteration Count =

3 3 3 3 33 3 3 3 33 3 3 3 33 3 3 3 33 3 3 3 3


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 1.09252.3095 2.1865 1.8378 1.2808 0.73863

0.85122 1.7251 1.6194 1.1398 0.621092.3095 2.1865 1.8378 1.2808 0.738634.687 3.399 2.51 1.7291 1.0925

Iteration Count =

3 3 3 4 43 3 4 4 44 4 4 4 43 3 4 4 43 3 3 4 4


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 + 0.50i14.13− 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 2.55085.3757 3.0210 1.9569 1.3639 1.48591.1754 1.1761 0.9225 0.3009 1.11425.3757 3.0210 1.9569 1.3639 1.4992

20.7682 9.7737 5.5177 3.5433 2.5508

Iteration Count =

3 3 3 4 43 3 5 5 55 5 5 5 53 3 5 5 53 3 3 4 4


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− 0.40i1.70− 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 0.751.70 + 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

z50 =

NaN = Not a Number Just too big!NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNiNaN + 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 + NaNiNaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi NaN + NaNi


Iteration Chaos

The bounded values are often different for each iteration. The followingvalues 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 −1.50z101 : 0.26748 0.7738 −1.6092 −0.36454 −0.24833z102 : −1.8285 −1.2012 0.88944 −1.4671 −1.4383z103 : 1.4432 −0.35703 −0.90889 0.55241 0.56879z104 : 0.18297 −1.6725 −0.87392 −1.2948 −1.1765z105 : −1.8665 0.99735 −0.93627 0.076621 −0.11591z106 : 1.5839 −0.8053 −0.82339 −1.5941 −1.4866z107 : 0.60876 −1.1515 −1.022 0.94125 0.70987

IterationCount at z63 3 3 4 43 3 5 5 66 6 6 6 63 3 5 5 63 3 3 4 4

IterationCount at z107

3 3 3 4 43 3 5 5 6

107 107 107 107 1073 3 5 5 63 3 3 4 4


Magnitude Contrasts at z5 and z7

Magnitude at z5 =20.7682 9.7737 5.5177 3.5433 2.55085.3757 3.0210 1.9569 1.3639 1.48591.1754 1.1761 0.9225 0.3009 1.11425.3757 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 26.384940.55 56.91 19.477 9.0814 3.40921.6313 1.626 0.9790 0.6784 1.4332940.55 56.91 19.477 9.0814 3.4092

1.8633e + 005 8826.5 1016.9 173.28 26.384


The Mandelbrot When Viewed Sideways

Figure: Mandelbrot Fractal SideImage

Here we see the Mandelbrotfrom a different perspective.The structure of the fringesis clearly seen

The top areas are where thevalues are bounded

The lower areas are wherethe values are unbounded


A Mandelbrot Spiral

Figure: The Spinners


Mandelbrot Snowflakes

Figure: Stuck Snowflakes


The Mandelbrot Within the Mandelbrot

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


The −π Multibrot Fractal

Figure: Mandelbrot Fractal

zn+1 = z−πn + c

All real numberscan be anexponent in theequationzn+1 = zx

n + c ,to create a“Multibrot,”x 6= 2.

Onlyzn+1 = z2

n + c isa Mandelbrot


Common Alternate Multibrots

Figure: zn+1 = z4n + c Figure: zn+1 = z7

n + c


Bibliography

Arnold, Dave. Writing Scientific Papers in LATEX

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: TheMathematics Behind the Computer Graphics Proceedings ofSymposia in Applied Mathematics, American MathmaticalSociety, Volume 39, 1980

Gratzer, George. More Math Into LATEX, 4th ed. Springer, 2007

Lamport, Leslie. LATEX A Document Preparation System, 2nded. Addison Wesley, 1985


the mandelbrot fractal: an imaginary journey · 2010-12-20 · figure:mandelbrot fractal of all...

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