The Challenge: Today’s math art challenge is to play the chaos game, and was inspired while I was perusing the excellent Power in Numbers by Talithia Williams. Her chapter on Fern Hunt indicated one of her research interests as Chaos Theory – something that always grabs kids attention, and leads to one of the more fascinating probability related math-art creations.
Materials Needed: randomizer (die, coin, etc.), paper, pencil, possibly graphing software.
Math concepts you could explore with this challenge: fractals, probability, randomness
Dr Hunt’s main focus was in “dynamical systems and applied probability”. Nevertheless, the connection to chaos is enough to open the door for a probability lesson where you can teach about Dr. Hunt and her work before leading into the following activity.
- Draw a triangle.
- Number each corner such that rolling a die indicates them each equally. (I numbered mine with {1,2} {3,4} and {5,6} on each corner so each vertex had a 1/3 probability of being selected with a fair 6-sided die roll.)
- Place a point in the triangle randomly.
- Randomly select a vertex (I rolled a die for this) and using a ruler, mark a point exactly halfway from your original point to that vertex.
- Repeat…. MANY times.
Ideally, this is an excellent activity to do on transparency paper in a class, and then overlay everyone’s images. The end result is rather surprising. Here is what I was able to do on my own (with about 2 hours of podcasts).
Simply because it takes a long time for the image to resolve itself if you’re doing this by hand, this may be a good one to try to code (not a current skill set of mine). This numberphile video does a nice job explaining and unpacking the chaos game, but if you start it where I have tagged it below (3:10) you can see the form taking shape:
Questions you might ask yourself or students participating in this:
- Can you predict the shape that will form?
- Were you surprised by what appeared?
- What is surprising and what may not be surprising about the final image?
- What about the “rules” for creating this imposed structure here?
- How might you alter the rules to get different results? (The wikipedia page actually explores some neat answers to that.)
- What do you imagine are actual applications for “applied probability”?
- What do you notice or wonder about the accuracy of by hand (mine above or yours created) and computer generated images? Is there a boon to one over the other? When may “by hand” be more useful? What about “computer generated”?
Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!
2.) Reason abstractly and quantitatively. How is “chaos” apparent here or not? How many trials do you need to make in order to see the pattern appear?
6.) Attend to precision. How is the effectiveness of the result affected by your precision in measuring?
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