The Chaos Game

This is an activity that I'll do with my students once in a while.  It's called the Chaos game, and I hear of it years ago from Robert Devaney,  back when the technology in my classroom consisted of an overhead projector.  So, this is my overhead projector version.  The same activity can be done very quickly using computers, but there's something to be said for hands-on point plotting.

Here's a link to the sheets I print out onto transparency paper.  (They're a little fuzzy, as they are scans of transparencies.)

Here's how it goes.

1.  Students are put in groups of three.  Each group is given a fine point maker (I prefer permanent in this case), a transparency sheet with the three vertices of an equilateral triangle on it, a plain white sheet of backing paper and a die.



2.  Students are instructed to choose a point at random within the triangle and to simply keep track of it with a pencil point, trying not to mark the transparency.  Let's call this point P₀. (In truth, the point can be chosen anywhere on the infinite plane, but to have the result emerge more quickly, the point should be within the triangle.)

3.  Students roll the die.  If it is a 1 or a 2, they find the midpoint between P₀ and A. If they roll a 3 or a 4, they find the midpoint between P₀ and B.  If they roll a 5 or a 6, they find the midpoint between P₀ and C.  They move their pencil tip to the midpoint they found.  Let's call that point P₁.

4.  They repeat step three, rolling the die and finding the midpoint between P₁ and the selected vertex.

5.  Keep repeating.  Once they have found P₄, they start marking successive points with a small dot using the permanent marker.

6.  After about 15-20 points, they can stop.

7.  Here is one group's result:


What do you notice?  What do you wonder?

8.  Take the old overhead projector that you managed to scrape up from a closet somewhere, and hope the bulb hasn't failed since you checked it.  Place all the transparencies on the projector, lining up the vertices.  Here's one class's result:

Now what do you notice and wonder?

You can see the general shape of what the result is, but it's not crisp like on a computer.  Sometimes students will recognize the pattern, especially if they've seen the Sierpinski triangle before.

9.  Now I place the pre-made Sierpinski triangle pile on top of their diagram.  I then remove the stages one-by-one.

You can see above that there are a lot of stray marks when you put them all together, and some of the points do sneak out into the smaller triangles, but given the number of points that are plotted, it's pretty impressive.  Peeling back the last Sierpinski triangle makes this clear.

That's it.  

The only question remains:  Why does it work?  I'll leave it to Robert Devaney to explain here.