Bruce Wayne Fractals

What is a Fractal?

According to Benoit Mandelbrot …

“A fractal is by definition is a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension.”

So … the concept of dimension is very important as we are learning about fractals.

Fractals in Nature

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

-- Benoit Mandelbrot

Advanced Synthetic Aperture Radar image of a

large glacial lake in Finland.

Fractal Clouds

Tajikistan

Fractal History

Helge Von Koch

Waclaw

Sierpinski

Georg Cantor

Gaston Julia

Benoit Mendelbrot

Richard Swearingen

Sunny

Dianne Clark

Fractal Terminology

Important Characteristics of Fractals

• They are recursive; that is, the process of their creation gets repeated indefinitely;

• They are self-similar; that is, copies of the entire fractal may be found, in reduced form, within the fractal.

Ways to Create Geometric Fractals

• use a base shape and replace it with a recurring motif shape (we did this when we created the Koch Triangle for homework, the initial triangle was the base and the shape that we replaced each side with was the motif)

• play the chaos game

• method of successive removals

Introducing

XAOS Software

Let’s look at this really neat fractal software and keep in mind those ideas about recursion and self-similarity.

Let’s Play …

The Chaos Game

Dimension

1 dimensional

2 dimensional

3 dimensional

Dimension Definition

(1) A measure of spatial extent, especially width, height, or length.

(2) The least number of independent coordinates required to specify uniquely the points in a space.

Dimension Definition

The first formal definition was stated by Dutch mathematician L E J Brouwer (1881-1966) in 1913.

“A (solid) cube has the topological dimension of three because in any decomposition of the cube into smaller bricks there always are points that belong to at least four (3+1) bricks.”

Definition

Self-similarity Dimension

D = log ( number of pieces )log ( magnification factor )

Easy example:

What is the self-similarity dimension of a cube that has a length = 3, a width = 3, and a height = 3 ?  

We can break the cube up into 27 smaller cubes, or "pieces".  Also, if we take one of the smaller cubes and "magnify" the sides by 3, we end up with a cube that is the same size as the original.  Hence, the "magnification factor" is 3.

Self-similarity dimension = log( number of pieces ) log( magnification factor )

Self-similarity dimension = log (27) = log(3)3 = 3 log(3) = 3 log (3) log(3) log(3)

What is the fractal dimension of the Koch Snowflake ?

Self-similarity dimension = log( number of pieces ) log( magnification factor )

What is the fractal dimension of the Koch Snowflake ?

Self-similarity dimension = log(4) = 1.26 log(3)

What would the "self-similarity dimension" be for the Koch Island Fractal ?

Self-similarity dimension = log ( number of pieces ) log ( magnification factor )

Self-similarity dimension = log (8) = log(2)3 = 3 log(2) = 1.5 log(4) log(2)2 = 2 log(2)

What is the area of the Koch Island fractal ?

What is the perimeter of the Koch Island fractal ?

Logo Programming

It is not just for the kids.

Big kids can have fun with it as well !!!

Fractals with Sketchpad

Now that you have experimented with creating the Hat Curve Fractal, it’s time to make your own. 

•Go to FILE, then DOCUMENT OPTIONS.•Choose the ADD PAGE tab, then BLANK PAGE.•Click on OK. Use the same procedure as you did for the Hat Curve Fractal to create your own. •Start with a horizontal line segment.•Decide upon a rule to use. Creativity counts here! For example, two rules you have seen are to replace the middle third of the segment with a triangle or with a square. Type your rule on your page with a text box. After you have created your fractal, copy and fill in the table below on sketchpad. Pick some convenient starting length for your segment (other than 1).  Stage 0 1 2 3 4 n

Length

           

Sierpinski Pyramid

Fractal Cards

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Fractals in the K-16

Curriculum

Fractal References

A Fractals Unit for Elementary and Middle School Students, by Cynthia Lanius, Rice University, http://math.rice.edu/~lanius/frac/index.html 1996-2007

Build a Sierpinski Pyramid, by Paul Kelly, Mathematics Teacher, 92, 384. 1999.

Chaos Game Applet by Trevor Stone:  http://trevorstone.org/applets/ChaosGame.html

Exploring Geometry, by Dan Bennett Emeryville, CA: Key Curriculum Press. 2002.

Fractal Cards: A Space for Exploration in Geometry and Discrete Mathematics. Simmt, Elaine & Davis, Brent Mathematics Teacher, 91, 102. 1998.

Fractals:  A Toolkit of Dynamics Activities, by Jonathon Choate, Robert Devaney, and Alice Foster, Key Curriculum Press, 1999

Fractint, a free fractal generator: http://spanky.triumf.ca/www/fractint/fractint.html

GNU XaoS, a free interactive fractal zoomer: at http://wmi.math.u-szeged.hu/xaos/doku.php

Interactivate Website by Shodor:  http://www.shodor.org/interactivate/activities/ MSWLogo software, free software download of setup kit from Softronics.com at http://www.softronix.com/logo.html

Pythagoras Plugged In. by Dan Bennett Emeryville, CA: Key Curriculum Press, 2003.

The Great Logo Adventure: Discovering Logo on and Off the Computer, by Jim Muller, Doone Pubns. 1998 (has CD) , free download of PDF file and CD files at http://www.softronix.com/logo.html

Turtle Geometry: The Computer as a Medium for Exploring Mathematics, by Harold Abelson and Andrea diSessa, MIT Press, 1981