Julia Set Explorer
For every point c in the complex plane, the map z → z² + c generates a distinct
dynamical system. The Julia set Jc is the boundary between orbits that escape and orbits that remain.
When c lies inside the Mandelbrot set, Jc is a connected fractal. When c
lies outside it, Jc shatters into a Cantor dust. Click the navigator to explore the correspondence.
Scroll to zoom. Click to centre.
Julia sets and the Mandelbrot set
Fix a complex parameter c and iterate the quadratic map fc(z) = z² + c
for every starting point z₀ in the complex plane. The filled Julia set Kc
is the set of z₀ whose orbits remain bounded; the Julia set Jc is its boundary.
The colour of each pixel encodes the escape time — how many iterations before |z| exceeds 2 — using smooth
(continuous) normalisation to eliminate integer banding.
The dichotomy theorem
Douady and Hubbard proved in 1982 that the filled Julia set Kc is either connected or
totally disconnected (a Cantor set), with no intermediate cases. The criterion is precisely whether the
critical point z = 0 escapes: if the orbit of 0 remains bounded, Kc is connected;
if it escapes, Kc is a Cantor dust. This is exactly the definition of the Mandelbrot set — so
c ∈ M ↔ Jc is connected. The Mandelbrot set is literally the map of which Julia sets are
connected and which have shattered.
Notable Julia sets
Basilica (c = −1): two symmetric lobes connected at a Siegel-disk fixed point — the simplest connected Julia set after the unit circle (c = 0). Douady rabbit (c ≈ −0.123 + 0.745i): three-lobed fractal, the filled Julia set of the period-3 bulb of the Mandelbrot set. Dendrite (c = i): a tree-like connected set, but with no interior — every point of the filled Julia set is on its boundary. Siegel disk (c ≈ −0.391 − 0.587i): contains a rotation domain (Siegel disk) where orbits circle quasi-periodically without converging. The boundary of this disk is a complicated fractal. Airplane (c ≈ −1.755): at the tip of the main period-3 component; highly elongated Julia set showing period-doubling cascades.
Smooth escape-time colouring
Raw iteration counts produce visible "level set" banding. The smooth (Hubbard–Douady) formula removes this:
where i is the first iteration at which |z| exceeds 2. The fractional part of ν varies continuously
between integer escape bands, giving the smooth gradient visible in the exterior. Hue cycles through the spectrum
every 40 units of ν; the dark interior points never escaped.