Whatís the point?
Chaos Theory is a quite complex concept in Mathematics that can be simplified all the way down to:† In the real world where things change all the time, sometimes some of these things are just random.† The first step toward actually looking at Chaos Theory is an understanding of fractals.† Fractals lend themselves very well to illustration on computers.† We are going to use the process called iteration to let the computer draw fractals for us.† The image of the fern that you see to the right is a classic example of an iterated fractal that is based in Chaos Theory.
Chaos Theory - ďChaos theory is the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systemsĒ .†
Fractal - A rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole.
Iteration - A computational procedure in which a cycle of operations is repeated, often to approximate the desired result more closely.
 Bourke, P. (1991).† An Introduction to Fractals.† http://astronomy.swin.edu.au/~pbourke/fractals/fracintro/
 Dictionary.com.† http://dictionary.reference.com/search?q=iteration
Questions to think about
What is Kochís Curve?† This curve was created by Fabian Helge von Koch.† This is sometimes referred to as the coastline problem because it could represent a sea shore.† Kochís Curve has infinite length, so does a coast.† Because if you increase the number of iterations of the curve you can find that the length goes on forever.† Likewise if you try to measure the length of a coast you will find that you can always change your measurement tool to find a more precise measurement and it will be infinite.
1) Why does the triangle CDE need to be an equilateral triangle?
2) What does the iteration tool do to the images?† In mathematical terms describe what happens when you iterate.
3) Describe the process to draw and iterate Kochís Curve.
4) What does this fractal look like?† Describe it in words and try to draw it here.
5) What happens when you move point A or B?
6) What happens when you move point C towards A?† Towards D?† Towards B?
7) What happens when you move point E towards A or B?† Can you describe the motion of point B?
Sierpińskiís Triangle begins with a simple triangle. It then constructs the midpoints of each segment of the triangle and creates another triangle inside of the first by connecting all the midpoints.† It doesnít look or sound all that interesting at first, but when you apply the iteration a number of times, what comes out is pretty neat.
1) What can you tell about the area of the new triangles being created?† How does it relate to the area of the original triangle?
2) If you increase the number of iterations is it possible for there to be an infinite number of triangles within Sierpińskiís Triangle?
3) What do you notice about the triangles created by the iteration?
4) Would you expect anything about the fractal, other than the general shape, to change if one of the vertices is changed? Why or why not?
5) How do the interior triangles relate to the exterior triangles? Is there anything that can be said about how each iteration relates to the previous iteration?
The Pythagorean Fractal
The fractal you are constructing comes from the Pythagorean Theorem.† By using this diagram and iterating it we can create an interesting picture.
1) Can you think of another way to construct a square using Geometerís Sketch Pad?† Explain your idea here.
2) What does the iteration look like?† Will the image look different when you change the number of iterations?
3) What can be said about the triangles being formed by the iterations?
4) Are the squares related to each other? If so, how? If not, why not?
5) Remember, the Pythagorean Theorem is . How does this fractal relate to the theorem?