Imagine you're blindly throwing darts at a dartboard. You'd expect the darts to land completely randomly, making a chaotic mess of holes all over the board. You definitely wouldn't expect the darts to magically form a perfect picture of a beautiful, infinitely repeating triangle! But in math, there's a trick where absolute randomness does exactly that. It's called the Chaos Game.

How to Play the Chaos Game

You can actually play this game yourself with a piece of paper, a pencil, and a die. Here's how it works:

1. Draw your target: Make three dots on your paper to form a large triangle. Label them 1, 2, and 3. (If you're using a standard six-sided die, you can say 1 and 2 mean corner 1, 3 and 4 mean corner 2, and 5 and 6 mean corner 3).
2. Pick a starting point: Put your pencil anywhere on the paper. Just make a dot.
3. Roll the die! Let's say you roll a number that corresponds to corner 2.
4. Jump halfway: Draw an imaginary line from your current dot to corner 2. Move your pencil exactly halfway along that line, and draw a new dot.
5. Keep going: Roll the die again. From your new dot, look at whichever corner the die tells you, and jump exactly halfway to that corner. Draw another dot. Repeat this forever!

Because you're rolling a die, you're constantly jumping in random directions. If you do this 10 or 20 times, it just looks like a random sprinkle of dots. But if you have the patience to do this a million times—or better yet, have a computer do it for you in a fraction of a second—something mind-blowing happens.

The Sierpinski Triangle Appears

Instead of filling the paper evenly like TV static, the dots start to form a pattern. Huge empty triangles appear in the middle where no dot ever lands. Then, smaller empty triangles appear inside the remaining spaces. Eventually, the dots perfectly trace out a famous shape called the Sierpinski Triangle.

The Sierpinski Triangle is a fractal. A fractal is a shape that looks the same no matter how far you zoom in. If you take one small corner of the Sierpinski Triangle and zoom in, it looks exactly like the whole massive triangle! It's infinitely detailed.

How Does Randomness Make a Perfect Pattern?

It feels like a magic trick. How can random choices create something so structured?

Think about the "jump halfway" rule. Every time you jump halfway to a corner, you are basically taking the entire picture you've drawn so far and shrinking it by 50% toward that corner. You are doing this for all three corners. It turns out that the Sierpinski Triangle is the only shape in the universe that is made up of three smaller, 50%-shrunk versions of itself!

In math, we call the Sierpinski Triangle an attractor for these rules. It's like a magnet. No matter where you start your first dot, the rules of the game constantly pull your dots closer and closer to that perfect fractal shape. The die rolls just make sure your pencil randomly visits every single part of the infinite shape, filling it in over time.

Try It Yourself!

You don't need a million-sided die or a lifetime of drawing dots to see this happen. You can watch the Chaos Game unfold right in your browser! Check out our interactive Chaos Game simulator.

Try changing the rules! What happens if you start with a square instead of a triangle? What if you jump 1/3 of the way instead of halfway? You can create completely different, beautiful fractals just by tweaking these simple rules.