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.