1. What do these pictures have in common? Write a complete sentence.
What are Fractals?
• A fractal is a rough or fragmented
geometric shape
– can be broken into parts
– each part is a smaller copy of the whole.
Self Similarity
• If a fractal’s parts are copies of the whole the
fractal is self similar.
• If any given part of the fractal is an exact replica
of the whole fractal, the fractal is strictly self
similar.
• A fractal is entirely self similar or strictly self
similar, it cannot be part one thing and part
another.
• Go to the website below, then return to this
PowerPoint.
http://astronomy.swin.edu.au/~pbourke/fractals/selfsimilar/
Traits of Fractals
• Self-similarity-smaller regions
resemble the entire diagram
when we zoom in on specific
areas
• “Strictly” self-similar fractalsmade of exact copies of the
original put together.
• “Recursively” self-similar have some of the same
shapes in smaller sections as
in the bigger sections.
To make a fractal
• Begin with a base shape and a basic change
Base shape
Basic change
• Make the change to each successive stage of the
fractal. Each new stage of the fractal is called an
iteration.
1st iteration
2nd iteration
Another example
Base shape
1st iteration
Basic change
2nd iteration
MORE EXAMPLES
For the next few examples, continue to press enter so the fractal is generated.
You will see the following.
1.
2.
3.
4.
Tree
Sierpinski’s Triangle
Dragon Curve
Koch’s Snowflake
Tree
Stage three
two
six
four
seven
five
one
Dragon Curve
Koch Snowflake
PART II: Fractal dimension
What is dimension? How do we assign dimension to an object?
When you see:
a train moving along railroad tracks,
1. In what dimension does it move?
PART II: Fractal dimension
What is dimension? How do we assign dimension to an object?
When you see:
a boat sailing on a lake,
2. In what dimension does it move?
PART II: Fractal dimension
What is dimension? How do we assign dimension to an object?
When you see:
a plane in the sky,
3. In what dimension does it move?
I. Take an unused piece of aluminum foil:
Fractal dimension
III. When you carefully
reopen the ball of foil,
what dimension has it
become?
What is it's dimension?
II. Now, crumple it up into a ball:
What is the dimension of the ball of foil?
PART II: Fractal dimension
4. What is the dimension of a fractal between? ________________
Fractal dimension reference
http://www.math.umass.edu/~mconnors/fractal/dimension/dim.html
Go to the above link for more information about fractal dimension.
PART III. Iteration and orbits
• Use the worksheet and the following link at the
same time.
http://aleph0.clarku.edu/~djoyce/julia/julia.html
For more details, see
http://www.jcu.edu/math/vignettes/population.htm
http://www.ies.co.jp/math/java/misc/chaosa/chaosa.html
PART IV: Major Types of Fractals
Julia
Mandelbrot
Sierpinski Triangle
Koch Snowflake
The Mandelbrot Set
• Probably the most well
known fractal is the
Mandelbrot Set.
• The Mandelbrot Set is a
group of complex points
that have a magnitude
limit of 2 when iterated in
zn+1= zn2 + c
• The Map is the graph of
the points tested
(the points in the black area
are within the Mandelbrot
set while the colored
points are not)
The Mandelbrot Graph
• Coloring of the points
tested for a
Mandelbrot set varies
• Arbitrary color
assignments based
on number of
iterations it takes for
the magnitude of a
point to become
larger than 2 are
used.
PART IV: The Mandelbrot Set
Use the below website to complete PART IV of
your worksheet along with your calculator.
http://www.geocities.com/CapeCanaveral/2854/
Gaston Julia
• Gaston Julia was one of the first to work with the
limits of Fractals. His question was based on
the bounds of fractals with a given C. He asked
for what values of Z does the equation stay
bounded. So to find a number that left the
equation bounded he fixed a value to C and so
created the instructions for making a Julia set of
numbers. First fix a value to C and then find all
Zs that leave Z2 + C bounded.
The Julia Set f(z) = z2 + c
The Julia Set f(z) = z2 + c
Go to the following website and read the complex number example.
http://aleph0.clarku.edu/~djoyce/julia/julia.html
If you want more information, you can read more about Julia Sets here
http://www.geocities.com/CapeCanaveral/2854/
and click on Julia Sets on the left hand side.
http://www.mcgoodwin.net/julia/juliajewels.html
Go to the following link and explore the sets
using the applets.
http://nlvm.usu.edu/en/nav/frames_asid_136_g_3_t_3.html?open=instructions
We hope you enjoyed learning about Fractals. Next, your group will explore a particular
fractal and teach your classmates.
References
• Some of the slides were part of previous
Honors Precalculus Classes at Hinsdale
South High School. In addition, several
websites have been used to help you
understand the concepts.