For a complex polynomial f_c(z) = z² + c, the Julia set J_c is the boundary between initial points z₀ that remain bounded under iteration (the filled Julia set K_c) and points that escape to infinity. Named after Gaston Julia (1918) and independently Pierre Fatou. For c near the Mandelbrot set boundary, Julia sets are intricately connected fractals; for c outside the Mandelbrot set, the Julia set is a totally disconnected Cantor dust.

What makes the Douady rabbit Julia set special?

The Douady rabbit (c ≈ −0.1226 + 0.7449i, or in the default preset c = −0.7269 + 0.1889i) is named for its rabbit-ear shape and is one of the most studied Julia sets. It lies near the period-3 bulb of the Mandelbrot set. The filled Julia set consists of a main body with infinitely many smaller budding copies — a consequence of the 3-cycle in the dynamics of f_c at that parameter value.

Julia Set Explorer

Vary the complex parameter c and watch an entirely different fractal universe unfold. Each point on the parameter plane gives a distinct Julia set — connected or Cantor dust, rabbit or dragon.

Parameter c

Re −0.7269

Im 0.1889

Parameter plane (Mandelbrot) — click to set c

Presets

Max Iterations

Iter 128

Colour Palette

Shortcuts
Drag → pan
Scroll / pinch → zoom
R → reset view
S → save PNG
+/− → zoom in/out

∞ Infinite Variety

Every complex number c gives a different Julia set. Move c by 0.001 and the topology can change completely — from a connected fractal filled with intricate spirals to an uncountable cloud of isolated points (Cantor dust).

🔗 Mandelbrot Connection

The Mandelbrot set M is the map of Julia sets: c ∈ M if and only if J_c is connected. Clicking inside M in the parameter plane gives a connected, intricate Julia set; clicking outside gives Cantor dust. The shapes of J_c mirror the local structure of M around c.

🎨 Smooth Colouring

The log-log formula t = n − log⊂2;(log⊂2;|z|) removes the discrete banding of integer escape time. The result is a continuous colouring that makes the fractal’s true gradient structure visible — no artificial steps or contour lines.

🔬 Fatou-Julia Theory

The complex plane is split into the Fatou set (stable dynamics — open) and the Julia set (chaotic dynamics — its boundary). The filled Julia set K_c (coloured black) is the basin of attraction of ∞-stable points. It was described by Julia and Fatou independently around 1918.

The Algorithm: Quadratic Iteration on ℂ

For a fixed complex parameter c = a + bi, define the map f_c(z) = z² + c. The filled Julia set K_c is the set of starting points z₀ for which the orbit z₀, f_c(z₀), f²_c(z₀), … remains bounded. The Julia set J_c = ∂K_c is the boundary.

z_{n+1} = z_n² + c (c fixed, z_0 = pixel position) |z_n| > 256 ⟹ orbit escapes to ∞ (outside K_c) |z_n| ≤ 256 always ⟹ z_0 ∈ K_c (painted black) Smooth escape-time colouring: t = n + 1 − log₂(log₂(|z_n|)) This removes banding and gives continuous colour gradients. (Hubbard–Douady formula, valid for bailout radius ≥ 2)

The Fundamental Dichotomy

The most important structural result in complex dynamics (Fatou–Julia, independently 1918–19) is:

Fundamental Dichotomy: c ∈ M ⟺ J_c is CONNECTED (one piece) c ∉ M ⟺ J_c is a CANTOR SET (totally disconnected dust) where M = { c ∈ ℂ : orbit of 0 under f_c is bounded } = Mandelbrot set. Proof sketch: The critical point 0 is the only finite critical point of f_c(z)=z²+c. If its orbit stays bounded → all Fatou components of K_c are connected. If its orbit escapes → all Fatou components collapse to points → Cantor dust.

Famous Julia Sets

Name	c value	Character
Douady Rabbit	−0.7269 + 0.1889i	3-cycle bulb; rabbit-ear connected set
Siegel Disk	−0.8 + 0.156i	Rotation domain (irrational rotation number)
Cauliflower	0.285 + 0.01i	Near period-1 parabolic; bumpy boundary
Dendrite	−0.4 + 0.6i	Parabolic filled Julia set; tree-like
Basilica	0 + 1.0i	Period-2 bulb; two-lobe connected set
San Marco	−0.75 + 0i	On real axis; period-2 parabolic
Fibonacci	−0.1226 + 0.7449i	Period-3 Douady rabbit (alternate coords)
Cantor Dust	c outside M	c = 0.5 + 0.5i: totally disconnected

Topological Properties

Julia sets exhibit astonishing topological variety depending on c. When c lies in the interior of a hyperbolic component of the Mandelbrot set, the Julia set is a Jordan curve (simple closed curve) and K_c is a full, simply-connected disc topologically. Near the boundary ∂M, Julia sets become fractal — their Hausdorff dimension approaches 2. Douady and Hubbard (1982) proved that both J_c and M are locally connected in the hyperbolic setting, and Yoccoz proved it for many non-hyperbolic parameters.

The Siegel disk (one of today’s presets) is a particularly remarkable topological feature: an open region of the Julia set’s complement on which f_c acts as an irrational rotation. The boundary of the Siegel disk is a Jordan curve, but can be extremely complicated — in severe cases (Cremer points) it is nowhere locally connected.

FAQ

What is a Julia set?

The boundary of the set of complex numbers that remain bounded under iteration of z → z² + c (with c fixed). Named after Gaston Julia and Pierre Fatou who independently classified complex dynamics around 1918–19. Each distinct value of c produces a completely different Julia set — from connected fractals to Cantor dust.

How does the Julia set relate to the Mandelbrot set?

The Mandelbrot set is the “map of Julia sets”: c ∈ M if and only if J_c is connected. Click inside the purple Mandelbrot region on the parameter plane to see a connected Julia set; click outside to see Cantor dust. The local geometry of M around c predicts the local structure of J_c.

What is smooth escape-time colouring?

Standard colouring uses the integer iteration count n when |z| > escape, giving visible bands. The smooth formula t = n + 1 − log⊂2;(log⊂2;|z|) gives a continuous real escape time, removing banding. The result is smooth colour gradients that accurately represent the density of iteration at each point.

What makes the Douady Rabbit special?

The Douady Rabbit (c ≈ −0.7269 + 0.1889i) lies near the period-3 bulb of the Mandelbrot set where f_c has a 3-cycle attractor. The filled Julia set has three main “lobes” arrayed like rabbit ears, with infinitely many smaller copies recursively budding from each. It is one of the most studied Julia sets in complex dynamics.