Mandelbrot Sets on YouTubeo Mandelbrot Set Zoom

o http://www.youtube.com/watch?v=gEw8xpb1aRA

The Mandelbrot SetHalley Newman

Terminolgyo Mandelbrot Seto Iterated Function

Example—Iterated Functiono Let f(z)=z2+c, c is arbitraryo Let c=2o Our first input will be c.

o f(2)=(2)2+2=6o f(6)=(6)2+2=38o f(38)=(38)2+2=1446o f(1446)=(1446)2+2=2090918

o General Rule = Exponential Growth

Between -2 and 1o Weird Stuff Happenso Example: Let c= 0.1o f(0.1) = (0.1)2+0.1= 0.11o f(0.11) = 0.1121o f(0.1121) = 0.11256641o f(0.11256641) = 0.112671196

o Notice: This will converge, but will never reach 1. This is an exception to the general rule.

o Numbers between -2 and 0 behave differently

o Negative * Negative = Positiveo The square of all negative numbers

is positive.o Adding a Negative = Subtraction

o The positive gain you get between -2 and 0 is diminished by adding back the original negative number

Exampleo Iterate -1 in f(z) = z2 + co f(-1) = 1 + (-1) = 0o f(0) = 0 + (-1) = -1o f(-1) = 1 + (-1) = 0o f(0) = 0 + (-1) = -1o f(-1) = 1 + (-1) = 0

o The outputs alternate between -1 and 0 forever.

Exampleo What about c = -2, iterate this

using f(z) = z2 + co f(-2) = (-2)2 + -2 = 2o f(2) = (2)2 + -2 = 2o f(2) = (2)2 + -2 = 2

o Special Case!!!

o Other values between -2 and 0 behave even more strangely.

o They have chaotic behavioro Hidden Patterno Difficult to guess what the output

sequence will look like

Complex Numberso Real + Imaginary = Complexo Each have their own unique

address on the complex plane.

o Let’s study the same function f(z) = z2 + c

o We understand what happens when we iterate this function with real numbers between -2 and 1

o What happens when we use complex numbers in this same area?

f(z) = z² + co Fix a c on the complex plane.o Plug c into the function

f(z) = z²+ c.o Ex.o Fix c = - 0.5 - 0.5io So f(c) = (-0.5-0.5i)² + (- 0.5 - 0.5i)

Iterate f(z) = z² + (- 0.5 - 0.5i)o f(- 0.5 – 0.5i) = (- 0.5 – 0.5i)² +

(- 0.5 – 0.5i)o = 0.5i + (- 0.5 – 0.5i)o = - 0.5o Ando f(- 0.5) = (- 0.5)² + (- 0.5 – 0.5i)o = - 0.25 + - 0.5i

Iterate f(z) = z² + (- 0.5 - 0.5i)Continuing, we get…-0.6875+-0.25i-0.8984375e-1+-0.15625i-0.51634216+-0.47192383i-0.45610287+-0.1265166e-1i-0.29213024+-0.48845908i-0.6532522+-0.21461266i-0.11932016+-0.21960761i-0.5339902+-0.44759277i

Iterate f(z) = z² + (- 0.5 - 0.5i)And Eventually…-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i

Iterate f(z) = z² + (- 0.5 - 0.5i)oSo this function stabilizes

to a single value…

o-0.40867701+-0.27512526i

GraphingoWhenever a function

stabilizes, we will color its associated number (on the complex plane) blue. If, however, the function doesn’t stabilize, we will color its associated point red.

ExampleoThe function of(z) = z² + (- 0.5 - 0.5i)oStabilized too-0.40867701+ -0.27512526ioSo we’ll color - 0.5 - 0.5i blueoThe function f(z) = z² + (0.5 + i)oDoes not stabilize at alloSo we’ll color 0.5 + i red

The Mandlebrot SetoFor every point, we

determine whether the associated function stabilizes or not. If it does, we paint that point blue and if not, we paint the point red.

Warning!!!oSome functions stabilize

to more than one point (they cycle!).

oEx.of(z) = z² + (- 1 + 0i)

stabilizes too-1 and 0

Warning!!!We will still color it

blue!

Filling it in…

Mandelbrot Setso How do we get all the neat

colors in a Mandelbrot set?

Mandelbrot Seto Technically defined

as the set of all numbers c that do not grow exponentially in the iterated function f(z) = z2 + c

o These numbers are typically represented in black

Iterate 1+1i in f(z) = z2 + co f(1+1i) = (1+1i)2 + 1+1io = (1+1i) (1+1i) + 1+1io = 1+1i+1i+-1+1+1io = 2i +1+1io = 1+3i

o This output falls outside our circle so we know that the input 1+1i is not in the set, we color it red.

o If we perform the first iteration of the function for every number within the circle, each number whose first iteration output escapes the circle will also be colored red.

o If we continue this process using green for the second iteration and blue for the third iteration, we get:

o The edge can never be fully realized because it would take and infinite number of iterations to draw the complete boundary

o If we could perform an infinite number of iterations, we could see the boundary of the Mandelbrot Set is infinitely long even though it is contained within a circle of radius two

o Lastly, to display the same information, any sequence of colors can be used (choice of color is arbitrary) unlike the Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations.

Example

Example

Example

Example

Mandelbrot Seto Zoom Applet

o http://math.hws.edu/xJava/MB/

Mandelbrot Sets on YouTubeo Mandelbrot Set Zoom

o http://www.youtube.com/watch?v=gEw8xpb1aRA

Questions