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 fractals- made 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

• Make the change to each successive stage of the fractal. Each new stage of the fractal is called an iteration.

Base shape Basic change

1st iteration 2nd iteration

Base shape Basic change

1st iteration 2nd iteration

Another example

MORE EXAMPLES

For the next few examples, continue to press enter so the fractal is generated. You will see the following.

1. Tree2. Sierpinski’s Triangle3. Dragon Curve4. Koch’s Snowflake

Tree

Stage oneStage twoStage threeStage fourStage fiveStage sixStage seven

Dragon Curve

Koch Snowflake

PART II: Fractal dimensionWhat 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 dimensionWhat is dimension? How do we assign dimension to an object?

When you see:

                                                 

2. In what dimension does it move?

a boat sailing on a lake,

                                                  

PART II: Fractal dimensionWhat is dimension? How do we assign dimension to an object?

When you see:

                                                 

3. In what dimension does it move?

a plane in the sky,

                                                   

Fractal dimensionI. Take an unused piece of aluminum foil:

                                                 

What is it's dimension?II. Now, crumple it up into a ball:

        

                                           What is the dimension of the ball of foil?

III. When you carefully reopen the ball of foil, what dimension has it become?

PART II: Fractal dimension

4. What is the dimension of a fractal between? ________________

Fractal dimension referencehttp://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, seehttp://www.jcu.edu/math/vignettes/population.htmhttp://www.ies.co.jp/math/java/misc/chaosa/chaosa.html

PART IV: Major Types of Fractals

Mandelbrot Julia

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.