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

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 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

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 f c : C → C where f ( z ) = z 2 + c . Step 2: For each w ∈ C , recursively define the sequence { w n } ∞ n =0 where w 0 = w and w n = f ( w n − 1 ) . The sequence w n ∞ n =0 is referred to as the orbit of w . Step 3: ”Collect” all the w ∈ C whose orbit is bounded, i.e., let K c = { w ∈ C : sup | w n | ≤ M , for some M > 0 } n ∈ N and let J c = δ ( K c ) where δ ( K ) is the boundary of K . J c is called a Julia set . Matt Ziemke Fractals and the Mandelbrot Set

Julia Fractals - Example Let c = 0 . 375 + i (0 . 335). Consider w = 0 . 1 i . Then, w 1 = f ( w 0 ) = f (0 . 1 i ) = (0 . 1 i ) = 0 . 365 + 0 . 335 i w 2 = f ( w 1 ) = f (0 . 365 + 0 . 335 i ) = 0 . 396 + 0 . 5796 i w 20 ≈ 0 . 014 + 0 . 026 i In fact, { w n } ∞ n =0 does not converge but it is bounded by 2. So 0 . 1 i ∈ K c . Consider x = 1. Then, x 1 ≈ 1 . 375 + 0 . 335 i x 2 ≈ 2 . 153 + 1 . 256 i x 3 ≈ 3 . 434 + 5 . 745 i x 4 ≈ − 20 . 843 + 39 . 794 i x 5 ≈ − 1148 . 782 − 1658 . 450 i So looks as though 1 / ∈ K c . 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 J c is connected if and only if the orbit of zero is bounded, i.e., if and only if 0 ∈ K c . Matt Ziemke Fractals and the Mandelbrot Set

The Mandelbrot Set cont. A natural question to ask is... What does M = { c ∈ C : J c is connected } = { c ∈ C : { f ( n ) (0) } ∞ n =0 is bounded } c 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 J c is totally disconnected then J c is homeomorphic to the Cantor set. 2.) f c : J c → J c 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. 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 ) (0) } ∞ n =1 is not c attracted to a cycle? 3.) Is µ ( δ ( M )) > 0? Where µ is the Lebesgue measure. Matt Ziemke Fractals and the Mandelbrot Set