RememberingBenoit

Mandelbrot

20 November 1924 – 14 October 2010

First Citizenof

Science

(1924 – 2010)

Fatherof

Fractal Geometry

(1924 – 2010)

Theoryof

Roughness

(1924 – 2010)

The FractalGeometryof Nature

1977

1982

1985

The year when I metBenoit Mandelbrot

andRichard F. Voss

December 6, 1982

Leo Kadanoff

University of Utah

Mandelbrot Set 1980

1986

The mathematics behindthe Mandelbrot Set

University of California at Santa Cruz, October 1987

1988

Publishing all the algorithms

known at that time

How Mountains turn into Clouds …

A Masterpiece by Richard F. Voss

A completely synthetic mathematical

construction of mountains and clouds

1991...

1991...MaletskyPerciante

Yunker

PeitgenJürgensSaupe

1992

Mandelbrot Set:

The most complex object mathematics has ever seen

Iteration

Iteration of rational functions

Theory of Julia & Fatou~1918

€

Choose z0 in the complex plane.

Then iterate, which means compute

zn+1 = f (zn ) for n = 0,1,2,3,...

€

f (z) =p(z)

q(z), where p(z) and q(z) are polynomials

€

Example : f (z) =2z3 +1

3z2

I studied thatin the fall of 1982

at the University of Utah

Newton's Method for x3-1

Julia Sets

€

Given a rational function f (z),

collect all starting points z for which the

iteration does not go to infinity

J = z | z→ f (z) → f ( f (z)) → ...{ → ∞}

"The iteration does not escape to infinity"

"The Prisoner Set"

€

The Set of Complex Numbers C

z = a+ bi, i = −1

€

Addition

z = a+ bi, w = c + di

z + w = (a+ c) + (b+ d)i

€

Multiplication

z = a+ bi, w = c + di

z • w = (a+ bi) + (c + di) = (ac −bd) + (ad + bc)i

a

b z

€

The Set of Complex Numbers C

z = a+ bi, i = −1

€

Division? Find inverse to z = a+ bi :

1

z=

1

a+ bi=

1

a+ bi•a−bi

a−bi=a−bi

a2 + b2

a

b z

1/z

€

The Set of Complex Numbers C

z = a+ bi, i = −1

€

Modulus

z = a+ bi

| z | = a2 + b2

a

b z

€

The Quadratic Family

f (z) = z2 + c, c ∈ C

€

z0,

z1 = z02 + c,

z2 = z12 + c = z0

2 + c( )2

+ c = z04 + 2cz0

2 + c 2 + c

z3 = z22 + c = z0

4 + 2cz02 + c 2 + c( )

2+ c = ...

Julia Set

€

Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}

i.e. choose c and then

collect all starting points for which the iteration

does not go to infinity (Prisoner Set)

Theorem of Julia & Fatou

€

Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}

is

- either one piece (connected)

- or an infinite dust (Cantor Set)

Theorem of Julia & Fatou

€

Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}

is connected if and only if

c → c 2 + c → (c 2 + c)2 + c → ... → ∞

connected not connected

dust

connected not connected

(super) infinite dust

Cantor Set

Two simple Julia Sets

€

Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}

€

c = 0 :

z→ z2 → z4 → z8 → ...

Two simple Julia Sets

€

Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}

1

€

z <1⇒ z2 = z2

= z • z < z

€

z >1⇒ z2 = z2

= z • z > z

€

c = 0 :

z→ z2 → z4 → z8 → ...

Two simple Julia Sets

€

Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}

1

€

c = 0 :

z→ z2 → z4 → z8 → ...

€

Is it connected? Need to check :

c → c 2 + c → (c 2 + c)2 + c → ...

€

c = 0, compute

c → c 2 + c → (c 2 + c)2 + c → ...

Two simple Julia Sets

€

Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}

+2

€

−2 ≤ z ≤ 2 ⇔ z ≤ 2

⇒ z2 − 2 ≤ 4 − 2 = 2

-2€

c = −2 :

z→ z2 − 2 → (z2 − 2)2 − 2 → ...

Two simple Julia Sets

€

Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}

+2-2€

c = −2 :

z→ z2 − 2 → (z2 − 2)2 − 2 → ...

€

Is it connected? Check for c = −2 :

c → c 2 + c → (c 2 + c)2 + 2 → ...

The Mandelbrot Set

€

M = c | Jc is connected{ }

⇔

M = c | c → c 2 + c → (c 2 + c)2 + c → ...→ ∞{ }

€

{−2,0}∈ M

The Mandelbrot Set

€

M = c | c → c 2 + c → (c 2 + c)2 + c → ...→ ∞{ }

sequence becomes unbounded"escapes"

sequence remains bounded"imprisoned"

Making a picture:(b/w)

1980

Computer (Pixel) Graphics

C64: 1982 16 colors

Macintosh: 1984 b/w--------------------------RGB 256x256x256only in few research labsUniversity of Utah

1/4-2

1

-1

The Mandelbrot Set

€

M = c | c → c 2 + c → (c 2 + c)2 + c → ...→ ∞{ }

all sequences become unbounded"escape"

some sequences remain bounded"imprisoned"

2

Making a picture:b/w

€

c 2 + c − c ≤ c 2 + c + c ⇒

€

c 2 + c ≥ c 2 − c

€

=c 2

€

When c > 2 then c → c 2 + 2 → c 2 + 2( )2

+ c → ... escapes

€

Whe need the Triangle Inequality :

a +b ≤ a + b

€

Whe will show :

c > 2⇒ c 2 + c > c

€

⇒ c 2 + c ≥ c 2 − c = c c −1( )

€

⇒ c 2 + c > c

The Mandelbrot Set

€

M = c | c → c 2 + c → (c 2 + c)2 + c → ...→ ∞{ }

"imprisoned"

2

"escapes"takes 5 steps to land

outside circle

"escapes"takes 13 steps to land

outside circle

Making a picture:(color)

1982/83Salt Lake City

Around the Mandelbrot Set

Powers of Ten

Similarity between

Julia Sets

and the

Mandelbrot Set

1/(period)2

Mandelbrot Set 1990 (Peitgen/Jürgens/Saupe)

Electrostatic Potential(key for mathematical understanding)

Flying the Mandelbrot Set

Interview Bremen1986

We will always remember