Fractals and the Mandelbrot Set Matt Ziemke October, 2012 Matt - - PowerPoint PPT Presentation

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Fractals and the Mandelbrot Set Matt Ziemke October, 2012 Matt - - PowerPoint PPT Presentation

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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

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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

Matt Ziemke Fractals and the Mandelbrot Set

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What is a Fractal?

”My personal feeling is that the definition of a ’fractal’ should be regarded 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 geometrical language. 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

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The Koch Curve- 10 Iterations

Matt Ziemke Fractals and the Mandelbrot Set

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5-Iterations

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The Minkowski Fractal- 5 Iterations

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5 Iterations

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5 Iterations

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8 Iterations

Matt Ziemke Fractals and the Mandelbrot Set

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Heighway’s Dragon

Matt Ziemke Fractals and the Mandelbrot Set

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Julia Fractal 1.1

Matt Ziemke Fractals and the Mandelbrot Set

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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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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 : sup

n∈N

|wn| ≤ M, for some M > 0} and let Jc = δ(Kc) where δ(K) is the boundary of K. Jc is called a Julia set.

Matt Ziemke Fractals and the Mandelbrot 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 + 0.335i w2 = f (w1) = f (0.365 + 0.335i) = 0.396 + 0.5796i w20 ≈ 0.014 + 0.026i In fact, {wn}∞

n=0 does not converge but it is bounded by 2. So

0.1i ∈ Kc. Consider x = 1. Then, x1 ≈ 1.375 + 0.335i x2 ≈ 2.153 + 1.256i x3 ≈ 3.434 + 5.745i x4 ≈ −20.843 + 39.794i x5 ≈ −1148.782 − 1658.450i So looks as though 1 / ∈ Kc.

Matt Ziemke Fractals and the Mandelbrot Set

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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?

Matt Ziemke Fractals and the Mandelbrot Set

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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

Matt Ziemke Fractals and the Mandelbrot Set

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c=-0.022803-0.672621i

Matt Ziemke Fractals and the Mandelbrot Set

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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., if and only if 0 ∈ Kc.

Matt Ziemke Fractals and the Mandelbrot Set

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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?

Matt Ziemke Fractals and the Mandelbrot Set

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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 the Cantor set. 2.) fc : Jc → Jc is chaotic. 3.) Julia fractals given by c-values in a given ”bulb” of M are homeomorphic. 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