The Chaos Game

Draw an equilateral triangle on a bit of paper. Draw it nice and big. Now pick a corner to start at. Next, get a die (a D-6) and assign two numbers to each corner. For example, the top corner can be 1 and 2, the left corner can be 3 and 4, and the right corner can be 5 and 6. Write these numbers near the corners on the outside of your triangle so you remember them.

Roll your die. The number you get tells you which corner you are heading towards. Find the midpoint of your current location and the corner you're heading towards. Use a ruler for this and try to be as accurate as you can. Mark this new point with a good-sized dot, that is your new location.

Repeat.

After about fifteen minutes your triangle should have lots of lovely dots, and at this stage you might even see a pattern emerging.

A pattern! (I hear you cry) How can there be a pattern, when I am moving randomly! Surely the dots will be scattered in a random manner, looking like the freckles on a pasty Irish face. But there is indeed a pattern. A very nice one in fact. A very familiar one, actually.

SPOILER ALERT

Please actually carry out this experiment before looking ahead.

I will now insert some line breaks to stop you seeing the pictures below. Don't scroll until you're ready.

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I didn't rub out my construction lines because I've been taught well :-)

Can you tell what it is yet?

Holy fractal, Batman! That there is Sierpinski's triangle! Or as my students would say: Illuminati confirmed.

OK, it doesn't look exactly like Sierpinski but it's definitely getting there. I need to do another thousand or so iterations.

I am a mathematician, which basically means I have turned being lazy into a career. So at this point I started thinking, why am I drawing this s*** when I could be running a simulation instead?

Here is the spreadsheet I used to simulate the Chaos game. It was kind of fun to set up, so I suggest you try it yourself before reading my formulae. You can probably find a much more elegant way to do it, but I'm a mathematician, dammit, not a programmer!

Teachers: this is a cool way to kill an hour with a group of students who know how to use a ruler and divide stuff by two. You could do it as I have suggested, with a triangle drawn on blank paper, but you could instead do it with a triangle drawn on a coordinate grid, with the corners having the coordinates I used in my spreadsheet. This way, students can practise finding the mid-points of two points (a skill that is needed for Higher tier GCSE and for AS Level). However, the numbers get nasty pretty quickly (as there is a root 3 involved). Even if you just use this as an exercise in measuring with a ruler, I think students can get a lot out of it. The pattern is so cool and unexpected, your students may even have that "wow" awe and wonder moment.

Have fun!

Emma x x x