This is a game designed to generate a two dimensional Sierpinski triangle (the actual dimension is approximately ). This is the introductory lesson for the Fractus Series.

An overhead projector, * plastic overlays , markers, 6 sided dice, rulers.

I The Box (puzzles)

Defining Chaos - Have the class each write down what they think chaos is; have them write a definition of chaos in their own words. Then come up with some universal definition amalgamating all of the definitions. Keep this written on the chalk board.

Inquire if anyone has heard of fractals or knows anything about them.

If yes, then have them talk of what they know.

Inform the class what they will be doing today. They will be generating what is called Sierpinski's triangle, and then talking about what they have done.

Have the class split up into groups of about 5 people each (depending on class size).

Hand out the plastic overlays, markers, and rulers. * The plastic overlays should have a large triangle already drawn on it about 6-7 inch sides (equilateral or other).

II The Game (on the plastic overlays)

1 - First pick three points at the vertices of the pre-drawn triangle. Label one of the vertices A(1,2), the second B(3,4), and the third C(5,6).

2 - Now start with any point in the triangle. This shall be our initial condition (IC), as we shall call it. Each group should pick a distinct initial condition.

3 - Now roll the die. Depending on what number comes up, move the IC half the distance to the vertex of the corresponding number, use a ruler for this. That is, if a 3 comes up, move the IC point half the distance to vertex B(3,4). Now erase the first point, that is, erase the IC and then begin again, using the result of the previous roll as the IC for the next roll. Roll the die again and move the new point one half the distance to its' respective vertex, and then erase the starting point. Continue this process for about 5 rolls.

4 - After these five rolls, begin to record the track of these traveling points after each roll of the die (do not connect them).

The goal of this game is to roll the die hundreds of times and predict what the resulting pattern of the points will be.

5 - Have the students try to guess what the picture will look like after having the groups plot points for about 10-15 minutes.

III The Mirror (reflections)

Most students not familiar with this game guess that the resulting image will be a random smear of points.

Others predict that the points will eventually fill in the entire triangle.

Both of these guesses are wrong.

Collect the plastic overlays and line up the triangles so they overlap corresponding vertices.

The resulting figure (probability 100%) will resemble what mathematicians call a Sierpinski triangle (denote this by S). Have S (picture below) ready to show the students what the picture would look like if the points on their markers was finer and they played the game for hundreds more iterations (use this word and define it for them telling them that is what they have been doing).

Inform them that this is one of the most basic types of geometric figures known as a fractal. Also inform them that no matter what their IC was, after many rolls (probability 100% as # rolls) the orbit of the point will fill out S.

Use the word orbit and define it (see definitions below). Alternatively, you may want to have the class attempt to define the above word.

1 - Come up with at least 5 different examples, in nature or everyday life, of re-iterations and orbits. Write these down.

Iteration (of a function): fnew (x) = f(fold (x)). In common parlance, plug the generated function value back into the function (this would count as one iteration.

Orbit (of a point): The path of a chosen function value (usually taken at discrete intervals).

Chaos Theory: A mathematical field of study which studies order underlying apparent disorder.