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

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