There have been many attempts to find three-dimensional fractals that are as beautiful, complex, and fascinating as the Mandelbrot set is in two dimensions. One approach has been to use hypercomplex numbers instead of complex numbers, such as the quaternion Julia set, but all variations of this fractal have the same sort of 'linked hoops' structure; the infinite variety of forms in the original Mandelbrot set just aren't there. (Although, the quaternion Julia set can still be quite beautiful!) Another approach is to stack slices of a 2D fractal on top of each other (as in my Julia's Eye
and Julia's Scaffold
pieces), but although this can create beautiful 3D structures, they aren't really 3D fractals.
uses spherical coordinates to create a 3D fractal that shows the same infinite detail and infinite variety of forms found in the original Mandelbrot set. This piece gives an overall view of an 8th-power Mandelbulb. In addition to the exterior form that is captured in this piece, there are many fascinating structures to be found by diving inside -- although I'm still figuring out how to make 3D-printable sculptures out of them! This piece is hollow, with a flat base with drain holes inside to minimize the amount of material used. It glows quite nicely if you place a small LED light inside through one of the holes.