Generalized Mandelbrot set

The generalized Mandelbrot set is an extension of the classical Mandelbrot set. Instead of the standard quadratic function $ f_c(z) = z^2 + c $, we consider a more general family of functions of the form: $$ f_c(z) = z^d + c, $$ where $ d \in \mathbb{Q} $ (a rational number), allowing for fractional exponents. The generalized Mandelbrot set is defined as the set of complex numbers $ c $ for which the sequence of iterates $ z_{n+1} = f_c(z_n) $, starting from $( z_0 = 0 )$, remains bounded.

To calculate the powers of complex numbers efficiently, polar coordinates are used, as the general power $ z^d $ can be expressed in polar form as: $$ z^d = |z|^d e^{i d \theta} = |z|^d (\cos(d \theta) + i \sin(d \theta)), $$ where $|z|$ is the modulus and $ \theta $ is the argument of $ z $. This representation allows easy computation of both the real and imaginary parts of $ z^d$: $$ \operatorname{Re}(z^d) = |z|^d \cos(d \theta), $$ $$ \operatorname{Im}(z^d) = |z|^d \sin(d \theta). $$ This approach enables the visualization of the generalized Mandelbrot set for any rational value of $ d $, producing intricate and diverse fractal structures.

Choose an iteration function for the generalized Mandelbrot set from the dropdown menu below to explore each generalization in detail.