Randomness and Chaos Games

How Simple Random Choices Produce Structured Patterns

The Chaos Game

The classic chaos game generates the Sierpinski triangle by repeatedly moving halfway toward a randomly chosen vertex of a triangle. Despite pure randomness, a highly ordered fractal appears.

R implementation

vertices <- matrix(c(0, 0, 1, 0, 0.5, sqrt(3) / 2), ncol = 2, byrow = TRUE)
n_points <- 40000
pts <- matrix(0, n_points, 2)
pts[1, ] <- c(0.5, 0.3)

for (i in 2:n_points) {
  v <- vertices[sample(1:3, 1), ]
  pts[i, ] <- (pts[i - 1, ] + v) / 2
}

df <- as.data.frame(pts[-c(1:100), ]) # drop transient
names(df) <- c("x", "y")

ggplot(df, aes(x, y)) +
  stat_bin2d(bins = 150) +
  scale_fill_viridis_c(option = "plasma", trans = "log10") +
  coord_fixed() +
  theme_void() +
  theme(legend.position = "none")

Chaos game on a triangle (Sierpinski) — density view

p5.js: live chaos game

Other Chaos Games

Square with restricted choices (Barnsley-like)

verts <- matrix(c(0, 0, 1, 0, 1, 1, 0, 1), ncol = 2, byrow = TRUE)
n_points <- 50000
pts <- matrix(0, n_points, 2)
pts[1, ] <- c(0.5, 0.5)
prev <- 1

for (i in 2:n_points) {
  choices <- setdiff(1:4, prev)
  v_idx <- sample(choices, 1)
  v <- verts[v_idx, ]
  pts[i, ] <- (pts[i - 1, ] + v) / 2
  prev <- v_idx
}

df <- as.data.frame(pts[-c(1:200), ])
names(df) <- c("x", "y")

ggplot(df, aes(x, y)) +
  stat_bin2d(bins = 120) +
  scale_fill_viridis_c(option = "magma", trans = "sqrt") +
  coord_fixed() +
  theme_void() +
  theme(legend.position = "none")

Chaos game on a square with a simple restriction — density view

p5.js: variable number of vertices

Key Observation

Randomness alone does not produce symmetry. The rules that restrict which random choices are allowed (or how the next point is computed) inject the symmetry. In the Sierpinski case the three vertices and the “halfway” rule together force the triangular symmetry we see.