A visualization of the Mandelbrot set, the most iconic fractal in mathematics. Each pixel represents a complex number c on the complex plane, colored by how quickly the iteration z(n+1) = z(n)^2 + c diverges. The boundary between convergent and divergent regions reveals infinitely complex, self-similar structure including the characteristic cardioid and period-2 bulb.
#' anyplot.ai
#' heatmap-mandelbrot: Mandelbrot Set Fractal Visualization
#' Library: ggplot2 3.5.1 | R 4.4.1
#' Quality: 85/100 | Created: 2026-05-30
library(ggplot2)
library(ragg)
# Theme tokens (Imprint palette)
THEME <- Sys.getenv("ANYPLOT_THEME", "light")
PAGE_BG <- if (THEME == "light") "#FAF8F1" else "#1A1A17"
ELEVATED_BG <- if (THEME == "light") "#FFFDF6" else "#242420"
INK <- if (THEME == "light") "#1A1A17" else "#F0EFE8"
INK_SOFT <- if (THEME == "light") "#4A4A44" else "#B8B7B0"
# Imprint sequential colormap for continuous escape iterations
SEQ_LOW <- "#009E73" # Imprint position 1 — brand green (boundary region)
SEQ_HIGH <- "#4467A3" # Imprint position 3 — blue (far exterior)
# Mandelbrot parameters
MAX_ITER <- 200
NX <- 800
NY <- 572 # maintains 3.5 : 2.5 complex-plane aspect ratio
x_seq <- seq(-2.5, 1.0, length.out = NX)
y_seq <- seq(-1.25, 1.25, length.out = NY)
# Build flat grid vectors (all NX * NY coordinate pairs)
real_vec <- rep(x_seq, times = NY)
imag_vec <- rep(y_seq, each = NX)
# Vectorized Mandelbrot iteration
# Narrows the active-index set each loop so work shrinks as points escape.
zr <- numeric(NX * NY)
zi <- numeric(NX * NY)
escape <- rep(NA_real_, NX * NY) # NA = inside the set
log_mag <- rep(NA_real_, NX * NY) # log2(log2(|z|)) at escape — for smooth coloring
for (i in seq_len(MAX_ITER)) {
idx <- which(is.na(escape))
if (length(idx) == 0L) break
zr_new <- zr[idx]^2 - zi[idx]^2 + real_vec[idx]
zi_new <- 2.0 * zr[idx] * zi[idx] + imag_vec[idx]
zr[idx] <- zr_new
zi[idx] <- zi_new
mod_sq <- zr[idx]^2 + zi[idx]^2
hit <- mod_sq > 4.0
if (any(hit)) {
hit_idx <- idx[hit]
escape[hit_idx] <- as.numeric(i)
# Smooth coloring: log2(0.5 * log2(|z|^2)) = log2(log2(|z|))
# Stable when mod_sq > 4 because 0.5 * log2(mod_sq) > 1 always holds.
log_mag[hit_idx] <- log2(0.5 * log2(mod_sq[hit]))
}
}
# Smooth escape count — eliminates integer banding at iteration boundaries
# escape + 1 - log2(log2(|z|)) : fractional part interpolates between iterations
escape_smooth <- escape + 1.0 - log_mag # NA (interior) passes through as NA
escape_vis <- sqrt(pmax(escape_smooth, 0.0)) # sqrt spreads boundary detail
df <- data.frame(
real = real_vec,
imag = imag_vec,
esc = escape_vis
)
TITLE <- "heatmap-mandelbrot · r · ggplot2 · anyplot.ai"
p <- ggplot(df, aes(x = real, y = imag, fill = esc)) +
geom_raster(interpolate = TRUE) +
scale_fill_gradient(
low = SEQ_LOW,
high = SEQ_HIGH,
na.value = "#050503", # fixed dark interior independent of theme
name = "Escape\ncount\n(√)",
guide = guide_colorbar(
title.position = "top",
barwidth = 0.7,
barheight = 8
)
) +
coord_fixed(ratio = 1) +
labs(
title = TITLE,
x = "Real axis",
y = "Imaginary axis"
) +
theme_minimal(base_size = 8) +
theme(
plot.background = element_rect(fill = PAGE_BG, color = PAGE_BG),
panel.background = element_rect(fill = PAGE_BG, color = NA),
panel.grid.major = element_blank(),
panel.grid.minor = element_blank(),
panel.border = element_blank(),
axis.title = element_text(color = INK, size = 10),
axis.text = element_text(color = INK_SOFT, size = 8),
plot.title = element_text(color = INK, size = 12, face = "bold"),
legend.background = element_rect(fill = ELEVATED_BG, color = INK_SOFT,
linewidth = 0.3),
legend.text = element_text(color = INK_SOFT, size = 8),
legend.title = element_text(color = INK, size = 10),
plot.margin = margin(12, 12, 12, 12)
)
# Save — square canvas: width = 6, height = 6, dpi = 400 → 2400 × 2400 px
ggsave(
filename = sprintf("plot-%s.png", THEME),
plot = p,
device = ragg::agg_png,
width = 6,
height = 6,
units = "in",
dpi = 400
)
Runnable source as JSON, for any HTTP client: https://api.anyplot.ai/specs/heatmap-mandelbrot/ggplot2/code. Any spec id and library id listed in llms-full.txt fit the same URL shape; every URL below is complete and callable.
{
"spec_id": "heatmap-mandelbrot",
"language": "r",
"library": "ggplot2",
"page": "https://anyplot.ai/heatmap-mandelbrot/r/ggplot2",
"hub": "https://anyplot.ai/heatmap-mandelbrot",
"code_json": "https://api.anyplot.ai/specs/heatmap-mandelbrot/ggplot2/code",
"spec_json": "https://api.anyplot.ai/specs/heatmap-mandelbrot",
"render_light_png": "https://storage.googleapis.com/anyplot-images/plots/heatmap-mandelbrot/r/ggplot2/plot-light.png",
"render_dark_png": "https://storage.googleapis.com/anyplot-images/plots/heatmap-mandelbrot/r/ggplot2/plot-dark.png",
"quality_score": 85.0,
"license": "MIT",
"guide": "https://anyplot.ai/llms.txt"
}Part of Mandelbrot Set Fractal Visualization on anyplot.ai.