Remembering Benoit Mandelbrot
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RememberingBenoit
Mandelbrot
20 November 1924 – 14 October 2010
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First Citizenof
Science
(1924 – 2010)
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Fatherof
Fractal Geometry
(1924 – 2010)
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Theoryof
Roughness
(1924 – 2010)
The FractalGeometryof Nature
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1977
1982
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1985
The year when I metBenoit Mandelbrot
andRichard F. Voss
December 6, 1982
Leo Kadanoff
University of Utah
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Mandelbrot Set 1980
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1986
The mathematics behindthe Mandelbrot Set
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University of California at Santa Cruz, October 1987
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1988
Publishing all the algorithms
known at that time
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How Mountains turn into Clouds …
A Masterpiece by Richard F. Voss
A completely synthetic mathematical
construction of mountains and clouds
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1991...
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1991...MaletskyPerciante
Yunker
PeitgenJürgensSaupe
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1992
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Mandelbrot Set:
The most complex object mathematics has ever seen
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Iteration
Iteration of rational functions
Theory of Julia & Fatou~1918
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€
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
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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"
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€
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
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€
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
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€
The Set of Complex Numbers C
z = a+ bi, i = −1
€
Modulus
z = a+ bi
| z | = a2 + b2
a
b z
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€
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 = ...
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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)
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Theorem of Julia & Fatou
€
Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}
is
- either one piece (connected)
- or an infinite dust (Cantor Set)
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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 → ... → ∞
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connected not connected
dust
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connected not connected
(super) infinite dust
Cantor Set
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Two simple Julia Sets
€
Jc = z | z→ z2 + c → (z2 + c)2 + c → ...{ → ∞}
€
c = 0 :
z→ z2 → z4 → z8 → ...
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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 → ...
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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 → ...
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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 → ...
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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 → ...
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The Mandelbrot Set
€
M = c | Jc is connected{ }
⇔
M = c | c → c 2 + c → (c 2 + c)2 + c → ...→ ∞{ }
€
{−2,0}∈ M
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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
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1/4-2
1
-1
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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
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€
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
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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
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Around the Mandelbrot Set
Powers of Ten
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Similarity between
Julia Sets
and the
Mandelbrot Set
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1/(period)2
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Mandelbrot Set 1990 (Peitgen/Jürgens/Saupe)
Electrostatic Potential(key for mathematical understanding)
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Flying the Mandelbrot Set
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Interview Bremen1986
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We will always remember