Take a look at this 3D rendering of Mandelbrot set, slightly modified to work in 3D spherical coordinates instead of 2D polar.

The story start around 20 years ago with a guy named Rudy Rucker, an American mathematician, computer scientist and science fiction author (and in fact one of the founders of the cyberpunk science-fiction movement). Back then of course, the hardware was barely up to the task of rendering the 2D Mandelbrot, let alone the 3D version – which would require billions of calculations to see the results, making research in the area a painstaking process to say the least.

So the idea slumbered for 20 years until around 2007 when the researcher, Daniel White, independently pictured the same concept and published the formula for the first time in November 2007 at the fractalforums.com web site. The basic idea is that instead of rotating around a circle (complex multiplication), as in the normal 2D Mandelbrot, we rotate around phi and theta in 3 dimensional spherical coordinates (see here for details). In theory, this could theoretically produce our amazing 3D Mandelbrot, but there was some mathematical problems to be solved. Then the work of other fractal explorers and mathematicians, like Paul Nylander and David Makin, gave life to the Mandelbulb, the 3D version of Mandelbrot set.

In this Daniel White’s page you can find the whole story, links to the mathematical concepts and many beautiful images.

See also this site to live explore the Mandelbulb set.