The Chaos Game

How does this work?

1. The Setup

We place 3 fixed points in a triangle (labeled A, B, C). We also put a "pen" at a random spot on the screen to start (the white dot).

2. The "Frog Jump" Rule

Every step, we do exactly the same thing -

1. Roll a die to pick one of the corners (A, B, or C) at random.
2. Jump exactly halfway towards that corner.
3. Draw a dot where you land.

Try the +1 Step button to see this visualized!

3. Why the shape appears? (The Secret)

You might think jumping randomly would create a messy cloud. But jumping halfway creates a "Forbidden Zone."

No matter where you start, you can never jump into the large upside-down triangle in the middle. By repeating this, you create smaller and smaller forbidden zones forever. This is called the Sierpiński triangle.

This creates a Fractal—a shape made of smaller copies of itself.

Generated by Gemini

Got it!