The chaos game: a fractal from chance
Name: Date:
Before the film: predict
Pick one of a triangle’s corners at random, jump halfway towards it, mark the spot and repeat thousands of times. Will you get a random cloud, or something else?
Starter
The corners are A (0, 0), B (8, 0) and C (4, 8). Start at (4, 4) and jump halfway to A, then C, then B, then C. Write down each point.
How do you find the point halfway between two points?
Core
Each stage of the Sierpiński triangle removes the middle quarter of every triangle. What share of the area is left after 1, 2 and 5 stages?
How many small triangles are there after 5 stages?
Each jump halves the distance to the fractal. After 10 jumps, what part of the starting distance is left?
Challenge
The Sierpiński triangle is made of 3 copies of itself at half the size. Its dimension d solves 2ᵈ = 3. Find d.
Play the game in a square with jumps of one half. Why do the points just fill the square?