Mandelbrot set

The Mandelbrot set is one of the most famous fractals, known for its intricate boundary. It is a set of complex numbers, which are numbers that can be written in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, with the property that $i^2 = -1$.

The set is defined by a simple iterative formula. For each complex number $c$, we generate a sequence of complex numbers $z_0, z_1, z_2, ...$ starting with $z_0 = 0$. Each next number in the sequence is calculated using the previous one:

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

If this sequence remains "bounded" (meaning the numbers in the sequence do not escape to infinity), then the complex number $c$ is part of the Mandelbrot set. In practice, we check if the magnitude of $z_n$ (its distance from the origin, $|z_n|$) exceeds a certain value, typically 2. If it stays within this boundary after many iterations, we consider $c$ to be in the set.

The colors in the visualization of the fractal usually represent how quickly the sequence for a given $c$ escapes to infinity. Points inside the set are often colored black.

Phoenix Fractal

The Phoenix fractal is a variation of the Mandelbrot set. The iteration formula introduces a "memory" effect, as the next state depends on the previous two states:

z_{n+1} = z_n^2 + c + P z_{n-1}

In this implementation, we are generating a Mandelbrot-like set. The point $c$ is the coordinate on the complex plane being tested, while $P$ is a fixed parameter that can be changed. The initial values are $z_0 = 0$ and $z_{-1} = 0$. When $P=0$, this is the standard Mandelbrot set.

For different values of the parameter $P$, we get different Phoenix sets.

Parameters

GUI Parameters

Complex Parameters (Control Points)