the infinitely complex… fractals jennifer chubb dean’s seminar november 14, 2006 sides available...
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The infinitely complex… Fractals
Jennifer Chubb
Dean’s Seminar
November 14, 2006Sides available at http://home.gwu.edu/~jchubb
Fractals are about all about infinity… The way they look, The way they’re created, The way we study and measure them…
underlying all of these are infinite processes.
Fractal Gallery
3-Dimensional Cantor Set
Fractal Gallery
Koch Snowflake
Animation
Fractal Gallery
Sierpinski’s Carpet
Menger Sponge
Fractal Gallery
The Julia Set
Fractal Gallery
The Mandelbrot Set
Dynamically Generated Fractals and Chaos
Chaotic Pendulumhttp://www.myphysicslab.com/pendulum2.html
Fractal Gallery
Henon Attractor
http://bill.srnr.arizona.edu/classes/195b/henon.htm
Fractal Gallery
Tinkerbell Attractor and basin of attraction
Fractal Gallery
Lorenz Attractor
Fractal Gallery
Rossler Attractor
Fractal Gallery
Wada Basin
Fractal Gallery
Fractal Gallery
Romanesco – a cross between
broccoli and cauliflower
What is a fractal?
Self similarityAs we blow up parts of the
picture, we see the same thing over and over again…
What is a fractal?
So, here’s another example of infinite self similarity…
and so on … But is this a fractal?
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
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.
Dimension
Topological Dimension 0
The Cantor Set(3D version as well)
Dimension
Topological Dimension 1
Koch Snowflake
Chaotic Pendulum, Henon, and Tinkerbell
attractors
Boundary of Mandelbrot Set
Dimension
Topological Dimension 2
Lorenz Attractor
Rossler Attractor
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 ε of each other.
This list is not exhaustive!
Box-counting dimension
Computing the box-counting dimension…
…
…
…
13log
3log
13093.19log
12log
17457.127log
48log
… and so on… 1.26186
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
Thank you!