Complex plane iterative fractal - z_{n+1} = z_n^2 + c
Julia sets are fractals defined by the iteration formula z_{n+1} = z_n^2 + c, where c is a complex constant. Unlike the Mandelbrot set (which varies c for z_0 = 0), Julia sets fix c and vary the starting point z_0 across the complex plane.
The Mandelbrot set serves as a "map" of all Julia sets. Points c inside the Mandelbrot set produce connected Julia sets, while points outside produce disconnected "Cantor dust" sets. The boundary of the Mandelbrot set contains the most interesting Julia sets with complex structures.
For each point z_0 in the complex plane, we iterate the formula. If |z_n| exceeds the escape radius (typically 2), the point "escapes" to infinity and is colored based on how many iterations it took. Points that never escape (remain bounded) form the Julia set and are colored black.