# Julia set

Julia set from (-.1, -.5) to (1, .1). Image credit: Andrew Cantino.

A Julia set is any of any infinite number of fractal sets
of points on the complex plane (see Argand
diagram) defined by a simple rule. Given two complex
numbers *z* and *c*, and the recursion *z*_{n+1} = *z*_{n}^{2} + *c*, the Julia set for
any given value of *c*, consists of all values of *z* for which *z*, when iterated in the equation above, does not "blow up" or tend
to infinity. Julia sets are closely related to the Mandelbrot
set, which is the set of all values of *c* for which *z* =
0 + 0*i* doesn't tend to infinity through application of the recursion.
The Mandelbrot set is, in a way, an index of all Julia sets, For any point
on the complex plane (which represents a value of *c*) a corresponding
Julia set can be drawn. We can imagine a movie of a point moving about the
complex plane with its corresponding Julia set. When the point lies inside
the Mandelbrot set the corresponding Julia set is topologically unified
or connected. As the point crosses the boundary of the Mandelbrot set, the
Julia set explodes into a cloud of disconnected points called **Fatou
dust**. If *c* is on the boundary of the Mandelbrot set, but
not a waist point, the Julia set of *c* looks like the Mandelbrot set
in sufficiently small neighborhoods of *c*.

Julia sets are named after the French mathematician Gaston Julia (1893–1978),
whose most famous work, *Memoire sur l'iteration des fonctions rationnelles*,
which provides the theory for Julia sets before computers were available
to computer or represent them, was written in a hospital in 1918 at the
age of twenty-five. As a soldier in World War I Julia had been severely
wounded and lost his nose. He wrote his greatest treatise between the painful
operations necessitated by his wounds.