The chaos game: a fractal from chance
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.
(2, 2), (3, 5), (5.5, 2.5), (4.75, 5.25).
How do you find the point halfway between two points?
Take the average of the x values and the average of the y values.
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?
3/4 = 75 %, 9/16 ≈ 56.3 %, (3/4)⁵ ≈ 23.7 %.
How many small triangles are there after 5 stages?
3⁵ = 243.
Each jump halves the distance to the fractal. After 10 jumps, what part of the starting distance is left?
(1/2)¹⁰ = 1/1,024, about a thousandth.
Challenge
The Sierpiński triangle is made of 3 copies of itself at half the size. Its dimension d solves 2ᵈ = 3. Find d.
d = log 3 ÷ log 2 ≈ 1.585: more than a line (1), less than a filled triangle (2).
Play the game in a square with jumps of one half. Why do the points just fill the square?
Four half-size copies of a square fill the square exactly, with no gaps, so the shape the rule leaves unchanged is the whole square.