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Fractals and the Mandelbrot Set
Matt Ziemke
October, 2012
Matt Ziemke Fractals and the Mandelbrot Set
Outline
1. Fractals
2. Julia Fractals
3. The Mandelbrot Set
4. Properties of the Mandelbrot Set
5. Open Questions
Matt Ziemke Fractals and the Mandelbrot Set
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.
Matt Ziemke Fractals and the Mandelbrot Set
The Koch Curve- 10 Iterations
Matt Ziemke Fractals and the Mandelbrot Set
5-Iterations
Matt Ziemke Fractals and the Mandelbrot Set
The Minkowski Fractal- 5 Iterations
Matt Ziemke Fractals and the Mandelbrot Set
5 Iterations
Matt Ziemke Fractals and the Mandelbrot Set
5 Iterations
Matt Ziemke Fractals and the Mandelbrot Set
8 Iterations
Matt Ziemke Fractals and the Mandelbrot Set
Heighway’s Dragon
Matt Ziemke Fractals and the Mandelbrot Set
Julia Fractal 1.1
Matt Ziemke Fractals and the Mandelbrot Set
Julia Fractal 1.2
Matt Ziemke Fractals and the Mandelbrot Set
Julia Fractal 1.3
Matt Ziemke Fractals and the Mandelbrot Set
Julia Fractal 1.4
Matt Ziemke Fractals and the Mandelbrot Set
Matt Ziemke Fractals and the Mandelbrot Set
Matt Ziemke Fractals and the Mandelbrot Set
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.
Matt Ziemke Fractals and the Mandelbrot Set
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 .
Matt Ziemke Fractals and the Mandelbrot Set
Julia Fractal - Example, Image 1
Matt Ziemke Fractals and the Mandelbrot Set
Julia Fractal - Example, Image 2
Matt Ziemke Fractals and the Mandelbrot Set
Julia Fractal - Example, Image 3
Why the colors?
Matt Ziemke Fractals and the Mandelbrot Set
c=-1.145+0.25i
Matt Ziemke Fractals and the Mandelbrot Set
c=-0.110339+0.887262i
Matt Ziemke Fractals and the Mandelbrot Set
c=0.06+0.72i
Matt Ziemke Fractals and the Mandelbrot Set
c=-0.022803-0.672621i
Matt Ziemke Fractals and the Mandelbrot Set
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 .
Matt Ziemke Fractals and the Mandelbrot Set
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?
Matt Ziemke Fractals and the Mandelbrot Set
The Mandelbrot Set cont.
Matt Ziemke Fractals and the Mandelbrot Set
The Mandelbrot Set cont.
Matt Ziemke Fractals and the Mandelbrot Set
The Mandelbrot Set cont.
Matt Ziemke Fractals and the Mandelbrot Set
The Mandelbrot Set cont.
Matt Ziemke Fractals and the Mandelbrot Set
The Mandelbrot Set cont.
Matt Ziemke Fractals and the Mandelbrot Set
The Mandelbrot Set cont.
Matt Ziemke Fractals and the Mandelbrot Set
M is a ”catalog” for the connected Julia sets.
Matt Ziemke Fractals and the Mandelbrot Set
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.
Matt Ziemke Fractals and the Mandelbrot Set
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
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