fractals and the mandelbrot set
TRANSCRIPT
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Fractals and the Mandelbrot Set
Matt Ziemke
October, 2012
Matt Ziemke Fractals and the Mandelbrot Set
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Outline
1. Fractals
2. Julia Fractals
3. The Mandelbrot Set
4. Properties of the Mandelbrot Set
5. Open Questions
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What is a Fractal?
”My personal feeling is that the definition of a ’fractal’ should beregarded in the same way as the biologist regards the definition of ’life’.”- Kenneth Falconer
Common Properties
1.) Detail on an arbitrarily small scale.2.) Too irregular to be described using traditional geometricallanguage.3.) In most cases, defined in a very simple way.4.) Often exibits some form of self-similarity.
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The Koch Curve- 10 Iterations
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5-Iterations
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The Minkowski Fractal- 5 Iterations
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5 Iterations
Matt Ziemke Fractals and the Mandelbrot Set
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5 Iterations
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8 Iterations
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Heighway’s Dragon
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Julia Fractal 1.1
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Julia Fractal 1.2
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Julia Fractal 1.3
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Julia Fractal 1.4
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Matt Ziemke Fractals and the Mandelbrot Set
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Julia Fractals
Step 1: Let fc : C→ C where f (z) = z2 + c .Step 2: For each w ∈ C, recursively define the sequence {wn}∞n=0
where w0 = w and wn = f (wn−1). The sequence wn∞n=0 is referred
to as the orbit of w.Step 3: ”Collect” all the w ∈ C whose orbit is bounded, i.e., let
Kc = {w ∈ C : supn∈N|wn| ≤ M, for some M > 0}
and let Jc = δ(Kc) where δ(K ) is the boundary of K . Jc is called aJulia set.
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Julia Fractals - Example
Let c = 0.375 + i(0.335).Consider w = 0.1i . Then,w1 = f (w0) = f (0.1i) = (0.1i) = 0.365 + 0.335iw2 = f (w1) = f (0.365 + 0.335i) = 0.396 + 0.5796iw20 ≈ 0.014 + 0.026iIn fact, {wn}∞n=0 does not converge but it is bounded by 2. So0.1i ∈ Kc .Consider x = 1. Then,x1 ≈ 1.375 + 0.335ix2 ≈ 2.153 + 1.256ix3 ≈ 3.434 + 5.745ix4 ≈ −20.843 + 39.794ix5 ≈ −1148.782− 1658.450iSo looks as though 1 /∈ Kc .
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Julia Fractal - Example, Image 1
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Julia Fractal - Example, Image 2
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Julia Fractal - Example, Image 3
Why the colors?
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c=-1.145+0.25i
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c=-0.110339+0.887262i
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c=0.06+0.72i
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c=-0.022803-0.672621i
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The Mandelbrot Set
Theorem of Julia and Fatou (1920)
Every Julia set is either connected or totally disconnected.
Brolin’s Theorem
Jc is connected if and only if the orbit of zero is bounded, i.e., ifand only if 0 ∈ Kc .
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The Mandelbrot Set cont.
A natural question to ask is...What does
M = {c ∈ C : Jc is connected } = {c ∈ C : {f (n)c (0)}∞n=0 is bounded}
look like?
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The Mandelbrot Set cont.
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The Mandelbrot Set cont.
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The Mandelbrot Set cont.
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The Mandelbrot Set cont.
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The Mandelbrot Set cont.
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The Mandelbrot Set cont.
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M is a ”catalog” for the connected Julia sets.
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Interesting Facts about M
1.)If Jc is totally disconnected then Jc is homeomorphic to theCantor set.2.) fc : Jc → Jc is chaotic.3.) Julia fractals given by c-values in a given ”bulb” of M arehomeomorphic.4.) M is compact.5.) The Hausdorff dimension of δ(M) is two.
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Open questions about M
1.) What’s the area of M?
2.) Are there any points c ∈ M so that {f (n)c (0)}∞n=1 is not
attracted to a cycle?3.) Is µ(δ(M)) > 0? Where µ is the Lebesgue measure.
Matt Ziemke Fractals and the Mandelbrot Set