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Plate 03 Guide

Julia set vs Mandelbrot set: same rule, different question

Julia sets and the Mandelbrot set use exactly the same formula. The difference is which number you hold still and which you vary.

A branching, lightning-like Julia set fractal in amber and cyan on a deep navy background

Same rule, different question

Both pictures repeat z squared plus c. For the Mandelbrot set, z always starts at 0 and each pixel is a different c. For a Julia set, c is fixed, the same for the whole picture, and each pixel is a different starting value of z.

So the Mandelbrot set is one map of all choices of c, and every Julia set is a picture for one single choice.

What a Julia set is

The Julia set for a given c is the border between starting points that stay bounded and starting points that escape. Julia sets are fractals, and they are named after the French mathematician Gaston Julia, who published a long paper on iterating functions in 1918. Pierre Fatou worked on the same questions at the same time, which is why the area is sometimes called Fatou and Julia theory.

The equation is the same z squared plus c. Only the roles of the numbers have changed.

How one point picks a Julia set

Every point c on the Mandelbrot plane has its own Julia set. Move c a little and the Julia set changes shape a little. This is exactly what the hero on the home page does: the position of your pointer is c, and the picture behind the headline is the Julia set for that c, redrawn every frame.

Connected or dust

The two kinds of picture are linked by a precise rule. If c is inside the Mandelbrot set, the Julia set for c is one connected piece. If c is outside, the Julia set shatters into disconnected points, often called dust. That is one reason the Mandelbrot set is described as an index of Julia sets.

Famous Julia sets

A few values of c have well known shapes:

  • c = 0: a perfect circle.
  • c = -2: a straight line segment.
  • c = -1: a double lobed shape often called the basilica.
  • c close to -0.123 + 0.745i: Douady's rabbit, with three ears.
  • c = i: a branching, lightning like dendrite.

Try it yourself

Any Julia set generator lets you type or drag c, and many 2D explorers show a Mandelbrot window next to a Julia window. The home page hero is a small example. For programs that do this at high resolution, see fractal art software.

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