🔬 Julia Set — Complex Dynamics

For each complex parameter c, the Julia set J_c is the boundary of the set of points z₀ for which the iteration z ↦ z² + c remains bounded. Click the left canvas (Mandelbrot set) to choose c — the corresponding Julia set appears instantly on the right. Points inside the Julia set are coloured by escape speed (iterations to diverge), revealing intricate self-similar structure. When c lies inside the Mandelbrot set, J_c is connected; outside it, J_c is a Cantor dust.

Mandelbrot set — click to set c

Julia set J_c

Parameter c

Re(c)-0.7269

Im(c)0.1889

Re(c) -0.73 Im(c) 0.19

Render

Max iterations 128 Zoom Julia 1.0×

Colour Map

Presets

Connected?—

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From Gaston Julia to Chaos Theory

Gaston Julia (1893–1978) studied iterated complex maps during World War I, while recovering from battle wounds, publishing his 199-page memoir in 1918. Before computers, these structures could only be imagined mathematically. Benoit Mandelbrot plotted the first images on an IBM computer in 1979, revealing the infinite, self-similar complexity now called the Mandelbrot set. A remarkable theorem by Douady and Hubbard (1985) proves that J_c is connected if and only if c belongs to the Mandelbrot set. The filled Julia set K_c is a compact set whose interior corresponds to bounded orbits; its boundary J_c is where chaotic behaviour lives — nearby initial conditions diverge exponentially, a hallmark of deterministic chaos.