Skip to main content

Randomness + Iterative Functions = Symmetry on Average

Symmetric Chaos and Density Patterns

The Central Idea

A single chaotic trajectory cannot be periodic, so it cannot possess a non-trivial symmetry.
Yet when we look at the density of a long orbit (or the attractor itself), clear symmetry often appears.
This is “symmetry on average”—the phenomenon explored in Field & Golubitsky’s Symmetric Chaos.

The mechanism is simple: iterate a map that commutes with a symmetry group (an equivariant map). The attractor is then invariant under that group.

Classic symmetric icon map (order 5)

# Field–Golubitsky style map tuned for visible structure (not a solid disk)
icon <- function(z, lambda = -1.8, alpha = 2.0, gamma = 1.0, m = 5) {
  r2 <- Mod(z)^2
  (lambda + alpha * r2) * z + gamma * Conj(z)^(m - 1)
}

n <- 150000
z <- 0.1 + 0.3i
pts <- complex(n)
for (i in 1:n) {
  z <- icon(z)
  pts[i] <- z
}
pts <- pts[-(1:2000)] # discard transient
pts <- pts[abs(pts) < 2] # keep only the attractor region

df <- data.frame(x = Re(pts), y = Im(pts))

ggplot(df, aes(x, y)) +
  stat_bin2d(bins = 80) +
  scale_fill_viridis_c(option = "inferno", trans = "log10", na.value = "black") +
  coord_fixed() +
  theme_void() +
  theme(
    legend.position = "none",
    plot.background = element_rect(fill = "black", color = NA)
  )

Density plot of a 5-fold symmetric chaotic attractor

Density plot of a 5-fold symmetric chaotic attractor

p5.js: interactive density accumulation

The sketch accumulates points; color cycles slowly. The five-fold symmetry emerges from the density.

Chaos Game as a Special Case

The chaos games of Document 2 are exactly this phenomenon for piecewise-linear equivariant maps.
The “average” (the histogram of visits) is the symmetric attractor.

Density version of the Sierpinski chaos game

vertices <- matrix(c(0, 0, 1, 0, 0.5, sqrt(3) / 2), ncol = 2, byrow = TRUE)
n <- 60000
pts <- matrix(0, n, 2)
pts[1, ] <- c(0.5, 0.3)
for (i in 2:n) {
  v <- vertices[sample(3, 1), ]
  pts[i, ] <- (pts[i - 1, ] + v) / 2
}
df <- as.data.frame(pts[-c(1:300), ])
names(df) <- c("x", "y")

ggplot(df, aes(x, y)) +
  stat_bin2d(bins = 90) +
  scale_fill_viridis_c(option = "magma", trans = "log10", na.value = "white") +
  coord_fixed() +
  theme_void() +
  theme(legend.position = "none")

Visit-frequency coloring of the chaos game

Visit-frequency coloring of the chaos game

Design Implications for Art Students

  • Equivariance is the design constraint: write (or choose) a rule that treats symmetric points the same way. The long-term picture will automatically be symmetric.
  • Color = density: the aesthetic decisions move to palette, contrast, and which regions you emphasize—exactly as in the Field–Golubitsky images.
  • Parameters are knobs: small changes in coefficients produce continuous families of related symmetric designs. This is parametric generative art with a mathematical guarantee of symmetry.
  • From instantaneous to average: the same idea appears in physical systems (Faraday waves, etc.). A single frame may look noisy; the time average reveals the symmetry.

Minimal working formula (copy-paste ready)

For cyclic symmetry of order ( m ):

[ z_{n+1} = (+ |z_n|^2) z_n + ^{m-1} ]

(with optional small imaginary term ( iz ) to break pure reflection symmetry and obtain pure rotational symmetry).

Iterate from a random start, discard the first thousand points, and color by visit count. That is the entire pipeline from the 1995 paper, reduced to a few lines of code.

Back to top