Fractals are a well-known corner of mathematics, and 2D ones are a common sight, but in 3D they're quite a bit harder to draw.  Strange attractors are a type that arise from relatively simple math: they're generated by iterative systems, in which equations are applied to their own results over and over again. The resulting path, instead of forming a simple shape, which is what you'd expect from simple equations, it skips around forever, slowly filling in a lacy fractal design.

Quaternion Julia sets have much in common with the more familiar 2D Julia sets, and indeed each contains a Julia set as its central cross-section.  A nice explanation of the math is here, and of course there are many more online.

This attractor was generated by Nicolas Desprez using his excellent freeware:

This crystal is a cube, 3 1/8" on a side.

Fractals are an extraordinarily interesting subject for this medum – one that demands a full-time specialist.  I have rendered a few others.