An infinite family of fractals — one for every point in the complex plane.
If the Mandelbrot set is a map of fractals, Julia sets are the fractals themselves. Every single point in the complex plane corresponds to a unique Julia set — an infinitely detailed shape that is either one connected piece or an infinite cloud of dust.
Julia sets use exactly the same formula as the Mandelbrot set:
The difference is what you fix and what you vary. In the Mandelbrot set, varies across the plane and is always the starting point. For a Julia set, you pick one fixed value of and then ask the escape question for every possible starting point .
Every complex number can serve as the constant in the Julia formula. Choose and you get a fractal dragon. Choose and you get a delicate web of tangent circles. Choose and you get a wild spiral archipelago.
Each coloured point in the complex plane defines a completely different Julia set.
There is a profound connection between a Julia set and the Mandelbrot set. Pick any value :
This means the Mandelbrot set is literally a parameter space: its shape encodes the topological type of every Julia set at once. Zoom into any feature of the Mandelbrot boundary and you're looking at a region where Julia sets transition from connected to disconnected.
The filled Julia set (also called the prisoner set) consists of all starting points that do not escape to infinity. The Julia set proper is the boundary of the filled Julia set — the infinitely thin frontier between escape and non-escape.
When you look at a Julia set render, you're usually seeing the filled Julia set coloured by escape speed, with the true Julia set being the razor-thin boundary between the coloured and black regions.
Like the Mandelbrot set, Julia sets are self-similar: zoom into any part of the boundary and you find structures that echo the whole shape. But each Julia set has its own characteristic repetition pattern, determined entirely by the single constant you chose.
Near (the tip of the left bulb of the Mandelbrot set) Julia sets develop long tentacle-like arms. Near (the tip of the antenna) the Julia set degenerates into a simple line segment: the interval on the real axis.
Quick fact
Gaston Julia discovered these sets in 1918 — without a computer. He worked out their properties entirely by hand, during and after recovering from severe wounds sustained in World War I.
The Julia Set Explorer lets you drag a point across the Mandelbrot set and watch the corresponding Julia set update in real time. Try clicking near the main cardioid boundary for connected, intricate shapes, or move outside it for shattered Cantor dust.
