The infinitely complex Fractals

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The infinitely complex Fractals

• Jennifer Chubb
• Deans Seminar
• November 14, 2006
• Sides available at http//home.gwu.edu/jchubb

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Fractals are about all about infinity

• The way they look,
• The way theyre created,
• The way we study and measure them
• underlying all of these are infinite processes.

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Fractal Gallery

• 3-Dimensional Cantor Set

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Fractal Gallery

• Koch Snowflake
• Animation

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Fractal Gallery

• Sierpinskis Carpet
• Menger Sponge

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Fractal Gallery

• The Julia Set

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Fractal Gallery

• The Mandelbrot Set

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Dynamically Generated Fractals and Chaos

• Chaotic Pendulum
• http//www.myphysicslab.com/pendulum2.html

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Fractal Gallery

• Henon Attractor
• http//bill.srnr.arizona.edu/classes/195b/henon.ht
  m

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Fractal Gallery

• Tinkerbell Attractor and basin of attraction

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Fractal Gallery

• Lorenz Attractor

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Fractal Gallery

• Rossler Attractor

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Fractal Gallery

• Wada Basin

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Fractal Gallery
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Fractal Gallery

• Romanesco
• a cross between broccoli and cauliflower

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What is a fractal?

• Self similarity
• As we blow up parts of the picture, we see the
  same thing over and over again

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What is a fractal?

• So, heres another example of infinite self
  similarity

and so on But is this a fractal?
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What is a fractal?

• No exact mathematical definition.
• Most agree a fractal is a geometric object that
  has most or all of the following properties
• Approximately self-similar
• Fine structure on arbitrarily small scales
• Not easily described in terms of familiar
  geometric language
• Has a simple and recursive definition
• Its fractal dimension exceeds its topological
  dimension

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Dimension

• Topological Dimension
• Points (or disconnected collections of them) have
  topological dimension 0.
• Lines and curves have topological dimension 1.
• 2-D things (think filled in square) have
  topological dimension 2.
• 3-D things (a solid cube) have topological
  dimension 3.

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Dimension

• Topological Dimension 0
• The Cantor Set
• (3D version as well)

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Dimension

• Topological Dimension 1
• Koch Snowflake
• Chaotic Pendulum, Henon, and Tinkerbell
• attractors
• Boundary of Mandelbrot Set

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Dimension

• Topological Dimension 2
• Lorenz Attractor
• Rossler Attractor

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Dimension

• What is fractal dimension?
• There are different kinds
• Hausdorff dimension how does the number of balls
  it takes to cover the fractal scale with the size
  of the balls?
• Box-counting dimension how does the number of
  boxes it takes to cover the fractal scale with
  the size of the boxes?
• Information dimension how does the average
  information needed to identify an occupied box
  scale?
• Correlation dimension calculated from the number
  of points used to generate the picture, and the
  number of pairs of points within a distance e of
  each other.
• This list is not exhaustive!

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Box-counting dimension

• Computing the box-counting dimension

• and so on 1.26186

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Hausdorff Dimension of some fractals

• Cantor Set 0.6309
• Henon Map 1.26
• Koch Snowflake 1.2619
• 2D Cantor Dust 1.2619
• Appolonian Gasket 1.3057
• Sierpinski Carpet 1.8928
• 3D Cantor Dust 1.8928
• Boundary of Mandelbrot Set 2 (!)
• Lorenz Attractor 2.06
• Menger Sponge 2.7268

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Thank you!