Last updated: 2026-09-26
U
Undergraduate levelMandelbrot Set: escape-time rendering with real-axis symmetry
For new readers
An interactive, zoomable Mandelbrot fractal renderer running live below, computed entirely by real PatLang code compiled to WebAssembly (not a JavaScript version with PatLang just narrating). Drag to pan, scroll to zoom. The second paragraph explains a specific speed optimisation (skipping half the work using the fractal's own mirror symmetry) — worth reading if you're curious why the timing readout changes as you pan, but not required to just play with the demo.
Overview & Architecture
A classic for a reason: an escape-time Mandelbrot renderer, compiled from real PatLang source to WebAssembly and run live below — every pixel's iteration count is genuine PatLang computation, not a JS reimplementation with PatLang narrating it. Pan by dragging the canvas; zoom by scrolling over it (zooms toward the cursor, not just the centre).
The one classic optimisation this demo doesn't skip: the Mandelbrot set is symmetric about the real axis — a point c and its complex conjugate always have identical escape behaviour, since squaring a complex number and its conjugate produces the same magnitude at every iteration. Whenever the current view is centred exactly on the real axis (cy == 0 — the default view, and where you land after "Reset view"), only the top half of the grid is actually iterated; the bottom half is copied directly from it, for roughly half the work. Panning away from the real axis breaks the exact pixel-for-pixel mirroring a screen row needs (row py's complex-plane offset and row h-1-py's are only exact negatives of each other when the view's own vertical centre is zero), so the fast path is strictly gated on cy == 0 rather than applied unconditionally when it would silently produce a wrong picture off-axis. Watch the timing readout roughly double the moment you pan off the real axis, and roughly halve again the moment you scroll back to a view centred on it.conjugate: same real part, imaginary part flipped
PatLang source
# Mandelbrot set escape-time renderer, with the classic real-axis
# (y -> -y) symmetry exploited to halve the work: c and its complex
# conjugate always have identical escape behaviour, so whenever the
# current view is centered on the real axis (cy == 0) only the top half
# of the grid is actually iterated -- the bottom half is a direct copy.
# Panning off the real axis (cy != 0) breaks that exact pixel-for-pixel
# mirroring (row py's y-coordinate and row (h-1-py)'s are no longer
# exact negatives of each other once the view's own center is offset),
# so the symmetry fast path is only gated on cy == 0 rather than
# applied unconditionally.
#
# argv: [cx, cy, scale, max_iter] (all optional, sane defaults below).
# scale is the complex-plane distance spanned by one pixel -- smaller
# scale = more zoomed in. Output: "GRID" then H lines of W single-char
# palette-bucketed pixels, then "TIME" and the render time in ms.
make a function called mandel_palette returns p
return "0123456789abcdefghijklmnopqrstuvwxyz"
end
make a function called mandel_bucket takes n, max_iter returns ch
if n >= max_iter then
return "."
else
let idx = to_num((n * 35) / max_iter)
return substr(mandel_palette(), idx, 1)
end
end
# Returns the escape-time iteration count for c = cx + cy*i (capped at
# max_iter for points that never escape the classic |z| > 2 bound).
make a function called mandel_iter takes cx, cy, max_iter returns n
let x = 0
let y = 0
let n = 0
let escaped = false
while (n < max_iter) and (not escaped) do
let x2 = x * x
let y2 = y * y
if x2 + y2 > 4 then
let escaped = true
else
let xt = x2 - y2 + cx
let yt = 2 * x * y + cy
let x = xt
let y = yt
let n = n + 1
end
end
return n
end
make a function called mandel_render_row takes cx, cy, scale, max_iter, w, h, py returns row
let b = sb_new()
# Pixel-CENTER sampling (py + 0.5, not py) is what makes row (h-1-py)
# an exact mirror of row py when cy == 0 -- it's what makes offset(py)
# and offset(h-1-py) exact negatives of each other for any h, not an
# approximation that happens to look right.
let y = cy + (py + 0.5 - h / 2) * scale
let px = 0
while px < w do
let x = cx + (px + 0.5 - w / 2) * scale
let n = mandel_iter(x, y, max_iter)
sb_push(b, mandel_bucket(n, max_iter))
let px = px + 1
end
return sb_str(b)
end
make a function called mandel_render takes cx, cy, scale, max_iter, w, h returns done
let use_symmetry = cy == 0
let half_h = h / 2
let rows = []
let py = 0
while py < h do
let need_compute = true
if use_symmetry and (py >= half_h) then
let need_compute = false
end
if need_compute then
let row = mandel_render_row(cx, cy, scale, max_iter, w, h, py)
let rows = list_push(rows, row)
else
let mirror_idx = h - 1 - py
let rows = list_push(rows, rows[mirror_idx])
end
let py = py + 1
end
print("GRID")
let i = 0
while i < h do
print(rows[i])
let i = i + 1
end
return true
end
let cli_args = argv()
let cx = -0.5
let cy = 0
let scale = 0.0125
let max_iter = 80
if cli_args.length > 0 then
let cx = to_num(cli_args[0])
end
if cli_args.length > 1 then
let cy = to_num(cli_args[1])
end
if cli_args.length > 2 then
let scale = to_num(cli_args[2])
end
if cli_args.length > 3 then
let max_iter = to_num(cli_args[3])
end
let t0 = now_ms()
mandel_render(cx, cy, scale, max_iter, 240, 240)
let t1 = now_ms()
print("TIME")
print(t1 - t0)
Interpreted run on the build machine (pat --ir-run, default view):
Output (click to expand)
GRID 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 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Max iterations: 80
View
Drag to pan. Scroll to zoom (toward the cursor). Symmetry engages automatically whenever the view is centred on the real axis.
Interactive Demonstration
starting...
Graduate Attributes
Modelled on this page
- Subject Specialist: Complex mathematical properties of the Mandelbrot set are applied to reduce computational workload by exploiting real-axis symmetry.
- Digitally Fluent: A custom language is compiled to WebAssembly to deliver a high-performance interactive visualisation directly in the browser.
- Reflective: The rationale behind a specific performance optimisation is explained alongside the implementation to clarify its impact on timing.
University of Reading Graduate Attributes framework.