Complex-Valued Fractals

The Mandelbrot Set

The Mandelbrot Set, defined and named by Adrien Douady and named for Benoit Mandelbrot. It is a color mapping of the complex plane, where every number is put repetitively through the equation $f_c(z) = z^2+c$ , with $c$ being the point that is to be tested and $z$ is initialized at 0. If this eventually breaks a limit, it will go to infinity, and it is therefore not in the set. In other words, if the sequence $f_c(0), f_c(f_c(0)),...$ ever goes beyond a given limit for a given $c$ , then that $c$ is not in the Mandelbrot Set. It is in the Mandelbrot Set, however, if it never breaks through the limit. Complex numbers in the Mandelbrot Set are colored black, and those that aren’t are colored based on how long it takes them to break the limit.

Julia Sets

Julia Sets are named after Gaston Julia. They are generated with the same equation as the Mandelbrot Set ( $f_c(z) = z^2 + c$ ), except instead $c$ is a parameter, and $z$ is every point on the complex plane. It is iterated and colored in the same fashion as the Mandelbrot set. Julia Sets aren’t restricted to the Mandelbrot Set (any function $f_c(z)$ will work), however it they are often referenced as so.

Burning Ship

The Burning Ship fractal was created by Otto Rössler and Michael Michelitsch. It is iterated and generated similarly to the Mandelbrot Set, but instead of $f_c(z) := z^2 + c$ , $f_c(z) = (|Re(z)| + i|Im(z)|)^2 + c$ . It is iterated and colored in the same way as the Mandelbrot Set. It shows chaotic areas when zooming in, such as warped mini-fractals and a white, chaotic net along the top (see left).

Rings Fractal

Inspired by Roger Bagula, Paul Bourke created the “Rings Fractal” which is generated very similarly to the rest of the escape-time fractals. It is described by the equation:

$z_{n+1}=\frac{z_n^2(z_n^2+\sigma)e^{2 \pi \phi i}}{\sigma z_n^2+1}$

Where $\phi = \frac{1+\sqrt{5}}{2}$

Nova Fractal

The Nova Fractal was described by Paul Derbyshire in the 1990s. It is generated using the equation:

$z_{n+1} = z_n - \frac{f(z_n)}{f'(z_n)}+c$

Where $f(z)$ is a function, generally $f(z) = z^3-1$ . $z_0$ is set to a critical point of $f(z)$ . It is usually rendered, not by the general escape-time method, but instead by how many iterations it takes for the difference in a value and its successor to become larger than a very small value (e.g., $10^{-12}$ ).