Mandelbrot/Phoenix — Fantastic Fractals

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

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's boundary but require more computation.

Complex Parameters (Control Points)

P: This is a fixed complex number that modifies the shape of the fractal. It can be moved around on the complex plane to explore different variations of the Phoenix set. When P=(0,0), this generates the classic Mandelbrot set.