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 )
3
Self-similarity dimension = log (27) = log(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)
log(3)
= 1.26
What would the "self-similarity dimension" be for
the Koch Island Fractal ?
Self-similarity dimension = log ( number of pieces )
log ( magnification factor )
3
Self-similarity dimension = log (8) = log(2) = 3 log(2) = 1.5
2
log(4) log(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
Length
0
1
2
3
4
n
Sierpinski Pyramid
Fractal Cards
crease
crease
crease
crease
crease
crease
crease
crease
crease
crease
crease
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