Here are some movies produced by John O'Connor displaying some self-similar fractals in three dimensional space. They are all attractors of iterated functions systems based on similarity transformations that map the unit cube onto four subcubes of sidelength one half with different orientations of the similiraties selected. They all have Hausdorff dimension 2. For some underlying theory see the paper 'Symmetry and enumeration of self-similar fractals' by K.J. Falconer and J. O'Connor, Bull. London Math. Soc., 39 (2007), 272-282.