This is an octahedron.

If you augment each face of an octahedron with more octahedra, you end up with this.

One can then augment each triangular face of this with yet more octahedra.

Here’s the next iteration:

This could, of course, go on forever, but one more step in the series is all you will see here. I don’t want to get caught in an infinite loop.

Performing various manipulations of polyhedra is easy with Stella 4d: Polyhedron Navigator, which I used to make all five of these rotating images. If you’d like to try this program for yourself, just check out http://www.software3d.com/Stella.php.