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!