When is the Julia set connected?

For z² + c, the filled Julia set is connected precisely when c lies in the Mandelbrot set (the critical orbit of 0 stays bounded). Outside, the Julia set breaks into a Cantor dust of points.

How is this different from the Mandelbrot set?

Mandelbrot varies c and always starts at z₀ = 0. A Julia image fixes one c and colors many starting z. The Mandelbrot set is the parameter map of which Julia sets are connected.

Why animate c on a circle?

A circular path in the c-plane periodically crosses the Mandelbrot boundary, so you see the Julia set morph between connected bulbs and dusty disconnected forms without manually hunting parameters.

Julia Set Explorer

Pick c by clicking the embedded mini-Mandelbrot or animate c along a circle; Fatou dust vs connected sets.

About this model

For a fixed complex parameter c, the Julia set of z ↦ z² + c is the boundary between orbits that escape to infinity and those that remain bounded (the filled Julia set / Fatou components). This explorer lets you pick c by clicking an embedded mini-Mandelbrot or animate c along a circle, switching between connected Julia sets (c inside the Mandelbrot set) and Cantor-like Fatou dust (c outside). Coloring follows escape time of the iteration z_{n+1} = z_n² + c from each pixel as initial z. The page focuses on the quadratic family only — no higher-degree maps or Newton basins — with finite iterations approximating true dynamics. Move or animate c to watch connectivity change.

Who it's for: Complex dynamics, fractal geometry, and undergrad courses linking Mandelbrot and Julia sets.

Key terms

Julia set
Filled Julia set
Fatou dust
Mandelbrot set
Complex quadratic map
Escape time

How it works

Julia set for a fixed parameter c: pixels are colored by how fast the orbit zₙ₊₁ = zₙ² + c escapes for that z₀. Picking c inside the Mandelbrot set gives a connected fractal; picking c outside gives a Cantor dust of disconnected points. Click on the small Mandelbrot map to pick c, or hit ▶ to walk c around the circle of radius 0.7885 — the Julia set morphs in real time.

Key equations

J(c) = ∂{ z₀ ∈ ℂ : supₙ|zₙ| < ∞ }, zₙ₊₁ = zₙ² + c

connected ⇔ c ∈ M (Fatou–Julia)

Frequently asked questions

When is the Julia set connected?: For z² + c, the filled Julia set is connected precisely when c lies in the Mandelbrot set (the critical orbit of 0 stays bounded). Outside, the Julia set breaks into a Cantor dust of points.
How is this different from the Mandelbrot set?: Mandelbrot varies c and always starts at z₀ = 0. A Julia image fixes one c and colors many starting z. The Mandelbrot set is the parameter map of which Julia sets are connected.
Why animate c on a circle?: A circular path in the c-plane periodically crosses the Mandelbrot boundary, so you see the Julia set morph between connected bulbs and dusty disconnected forms without manually hunting parameters.

Continue with similar topics in this category — or all 85 in Math Visualization.

View category →

NewSchool

Fractal Generator

Launch Simulator

Mandelbrot, Julia, Koch snowflake. Zoom infinitely.

NewUniversity / research

Mandelbrot Deep Zoom

Launch Simulator

Drag/wheel deep zoom into the Mandelbrot set with smooth continuous coloring and named landmarks.

NewUniversity / research

Newton-Raphson Basins in 2D Systems

Launch Simulator

Map Newton basins for nonlinear F(x,y)=0 systems: initial-guess sensitivity, iteration counts, root attraction, and singular-Jacobian failures.

NewSchool

Newton's Method (1D)

Launch Simulator

Graph f(x), click x₀, iterate x − f/f′ with numeric derivative; pan and presets. Complex basins: separate Newton fractal sim.

NewUniversity / research

Gaussian Process Regression

Launch Simulator

Interactive GP regression with RBF and Matérn kernels: tune length scale/noise, add observations by clicking, view posterior mean with uncertainty bands, and sample at max posterior variance.

NewSchool

Complex Phasor

Launch Simulator

exp(iωt) on the unit circle; Re, Im, and phase φ.

Julia Set Explorer

About this model

How it works

Key equations

Frequently asked questions

Related simulators

Fractal Generator

Mandelbrot Deep Zoom

Newton-Raphson Basins in 2D Systems

Newton's Method (1D)

Gaussian Process Regression

Complex Phasor

Julia Set Explorer

About this model

How it works

Key equations

Frequently asked questions