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Julia sets are close relatives of the Mandelbrot set. While the Mandelbrot iterates z \mapsto z^2 + c with z_0 = 0 and varies c across the image, a Julia set fixes c and varies the starting point z_0.
Each point c in the complex plane defines a unique Julia set. Points inside the Mandelbrot set produce connected Julia sets; points outside produce disconnected "Fatou dust."
The Mandelbrot set is essentially a map of all Julia sets: each pixel of the Mandelbrot tells you what the Julia set at that c value looks like:
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The parameter c completely determines the Julia set's geometry:
| c value | Character |
|---|---|
| c = 0 | Perfect circle (unit disk) |
| c = -1 | "Basilica", two tangent circles |
| c = -0.123 + 0.745i | "Douady's rabbit", three lobes |
| c = i | Dendrite, tree-like with no interior |
| c = -2 | Line segment [-2, 2] |
The boundary between connected and disconnected Julia sets traces out exactly the Mandelbrot set! Try picking a c value from the Mandelbrot notebook and rendering its Julia set here.