__What’s 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.

**Chaos Theory -** “Chaos
theory is the qualitative study of unstable aperiodic behavior in deterministic
nonlinear dynamical systems” [1].

**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.

[1] http://www.duke.edu/~mjd/chaos/chaos.html

[2] Bourke, P. (1991). An Introduction to Fractals. http://astronomy.swin.edu.au/~pbourke/fractals/fracintro/

[3] Dictionary.com. http://dictionary.reference.com/search?q=iteration

__Questions to think about__

*Koch’s
Curve*

What is Koch’s 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. Koch’s 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 Koch’s 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?

*Sierpiński’s
Triangle*

Sierpiński’s 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 doesn’t 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 Sierpiński’s 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 Geometer’s 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?