The Chaos Game

Problems

What would happen if we used just three vertices (a1, b1), (a2, b2), and (a3, b3)?

As with the square, we start with a point in the triangle. (In this example, it's on the edge of the triangle, but that's still in the triangle.)

Each move is half-way between where we are and a corner of the triangle, so we never leave the triangle.

Because we select the corners randomly, no part of the triangle is preferred over any other.

So since some parts of the triangle fill in, all parts must fill in.

Thus played with three vertices of a triangle, the chaos game should fill in the triangle. Right?

Here is the answer.

Perhaps we should rethink the reasons given at the end of Chaos Game Examples.

Here are four more Chaos Game examples. Try to determine the shape before running the program or looking at the answer.

Example 1: vertices the corners of a square, r = 1/3.

Example 2: move the top right vertex to the left, r = 1/2.

Example 3: five vertices, four the corners of a square, one at the center of the square, r = 1/2.

Example 4: five vertices, four the corners of a square, one at the center of the square, r = 1/3.

Return to the Chaos Game.