Of course, it is the Sierpinski tetrahedron! Lots of pictures of THIS fractal-of-the-Platonic-solids right on this site, and many other websites as well. It is a famous fractal.
The tetrahedron has four vertices, and the tetrahedron fractal grows in powers of 4. We have
seen only one deviance from this rule: the Menger Sponge, because the cube has only 8
vertices whereas the cube fractal grows in powers of 20.
More to come soon. This page has barely gone up on 27 May 2005.