Lab 7: The Chaos Game
On a piece of graph paper draw an isosceles right triangle and label the three vertices "1", "2", and "3". Take a six-sided die and identify the "1" and "2" with "1", the "3" and "4" with "2", and the "5" and "6" with "3", in order to generate a sequence of random numbers.
This game is called the chaos game. Begin by making a mark with your pencil anywhere on the paper. Roll the die and make your mark at the midpoint of the first mark and the vertex shown on the die. Continue rolling the die and marking the midpoint between your current position and the vertex shown on the die.
Below we have marked an initial point, which started outside the triangle, in red, and played four rounds of the chaos game. The four rolls were 2, 3, 2, and 1
What is the result of this game after many rounds? With games of 10, 50, or even 100 rounds, the result may not be clear ...
... but with games of 500, 1000, or 5000 rounds a pattern begins to emerge! How about 22,000 rounds?
