Do 3 Dimensional Fractals exist?

Question

I understand that certain mathematical sets produce fractals. Are there fractals defined by sets with more than 2 variables? Is that possible?

Answer 1

Yes, they do. Most fractal constructions can be taken into three dimensions. One of the easiest to visualize is to make a 3D version of the Sierpinski carpet.

Take a cube, cut it into $3 \times 3 \times 3$ subcubes, checkerboard color them with the corners black, and remove all the white ones. Now partition the subcubes you kept and keep going. The result is