Gasket--Attractors

Introduction

The concept of an attractor is important in many kinds of fractal objects. This lesson introduces this concept visually and experimentally, and suggests activities for exploring attractors using appropriate software.

Objectives

Students will learn what an attractor is
Students will learn how to use the computer application Gasket to explore fractal attractors
Students will be exposed to the following concepts:
- An iterative process
- An infininte process
- A limit
Students will use/learn about/rediscover the following tools
- Addition and multiplication of fractions
- Inductive reasoning
- Spatial relationships

Standards

These lesson ideas and activities address the following standards from the National Council of Teachers of Mathematics Curriculum Standards:

Overview

Students will first be introduced to the idea of an attractor with the square root function on a calculator, then with the "chaos game" on three points. They will see how to explore other attractors by changing the rules of the chaos game.

Materials

Actvities

Have the students do the following activity:

Draw a number line representing the positive real numbers from (just above) zero to any number (well, any number larger than two) you want. Label a few points on the line. Make the line as wide as the paper you are working with.
Pick any number on the line, and make a mark on that point.
Use a calculator to find the square root of that number, and make a mark on the corresponding location on your number line. Draw an arc connecting the first number you chose with its square root.
Now find the square root of the new number, plot it on the number line, and draw an arc connecting it to the previous number.
Repeat this process until successive points are too close together to be distinquished.
When the points are too close together, choose another starting point and repeat the process. This time, make the marks and the arcs in another color, or with dashed rather than solid lines, etc.

Ask the students what they notice about the patterns they have drawn. (They should notice that no matter what point they chose to start out with, they end up approaching 1.) The process of iteratively taking the square root of something is said to have an "attractor". The attractor is the number 1. They can further test this by taking the square root of any positive number iteratively until the calculator simply puts a 1 in the display.

THere are many iterative processes with attractors. Sometimes a process has more than one attractor. Computers use this type of behavior to do things like, well, calculating square roots. Often, when you ask the calculator to do something, it quickly goes through many steps of an iterative process and prints out the result. The calculator knows how to pick an iterative process that has the answer you want as an attractor.

Some processes have extremely complex attractors. Even a simple process can have an attractor that has an infinite number of points. One such process is the chaos game. (If the students have not seen the chaos game before, explain the rules and demonstrate the result.) The attractor for the standard chaos game is the Sierpinski Gasket. There are infinitely many points on the Sierpinski Gasket, but it is an attractor just like 1 for the repeated square root taking process. Notice the similarities:

You end up there no matter where you start
Once you're there, you never get away

If you take the square root of one, you get one. That's all you'll ever get, no matter how many times you do it. Similarly, if you pick any point on the Sierpinski Gasket, and move halfway toward any of the control points, you will be on another point of the Sierpinski gasket.

It is worthwhile to note that the experiments we have done don't prove that you will end up on the attractors for these two processes. Proving these results is probably more than we want to get into at this point. However, there are some things that are easy to understand and which can help the students see why each attractor attracts.

For the repeated square root, they can realize that
- Any number larger than one will give a bigger number when squared.
- Similarly, any number greater than zero and less than one gives a smaller number when squared.
- Since the square root is the "backwards" version of squaring a number, we can see that taking the square root of a number larger than one gives a smaller number, and that taking the square root of a positive number smaller than one gives a larger number.
We can also argue with these facts that repeatedly taking the square root, if starting with a number larger than one, will never give us a number smaller than one.
With the Sierpinski Gasket, it is easy to see that no matter where you start from, you will eventually end up inside the triangle defined by the three control points. Even if you started from very far away, all of your moves will take you toward the triangle. Eventually you will end up inside the triangle (unless your random number generator decides to quit choosing one of the points entirely).
Since you only move halfway to the selected point each time, you will never move back outside of the triangle once you are inside of it.

Next, have the students use Gasket to experiment with attractors for different setups than the standard chaos game. Have them change the the movement ratios and see if they can understand what is going on. See if they can figure out what the attractor looks like, or find settings that produce an interesting attractor. In particular, they should try using various numbers of points. Assign them to find an interesting attractor, tell why they find it interesting, and draw it freehand if possible.

The Big Picture

Thinking Harder

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