Julia — Fantastic Fractals

The Julia set is a fractal that is closely related to the Mandelbrot set. Like the Mandelbrot set, it is generated by iterating a simple formula on complex numbers.

The iteration formula for a Julia set is:

z_{n + 1} = z_{n}^{2} + c

Unlike the Mandelbrot set, where the initial $z_{0}$ is always 0 and $c$ varies over the plane, for a Julia set, $c$ is a fixed complex number. The initial value $z_{0}$ is the point on the complex plane being tested (i.e., the coordinate of each pixel).

If the sequence of $z_{n}$ for a given $z_{0}$ remains bounded (does not escape to infinity), then $z_{0}$ is part of the Julia set for that specific $c$ . Different values of $c$ produce vastly different Julia sets. Some are connected, while others are disconnected dust-like structures.

The Mandelbrot set can be thought of as a "map" of all possible Julia sets. If the value of $c$ is chosen from within the Mandelbrot set, the corresponding Julia set will be connected. If $c$ is outside the Mandelbrot set, the Julia set will be a disconnected set of points.

Parameters

GUI Parameters

Iterations: This integer value determines the maximum number of times the formula is applied for each point. Higher values increase the detail and accuracy of the fractal.

Complex Parameters (Control Points)

C: This is a fixed complex number that defines the specific Julia set to be rendered. It can be moved around on the complex plane to explore different Julia sets. It corresponds to $c$ in the iteration formula.