Recursive 3D Tiling of Rhombic Dodecahedra
I have a question about rhombic dodecahedra - polyhedra which can be tiled indefinitely, and which correspond to the closest packing of spheres. A single rhombic dodecahedron can be surrounded by 12 rhombic dodecahedra, making a tiling of 13 (having no gaps or holes or cracks). This creates a new volume which can be described as the union of 13 rhombic dodecahedra.
My question is this: Can I then take 13 of these shapes and tile them in the same fashion, as if they were each rhombic dodecahedra? And can this process be repeated indefinitely? If so, the boundary would approximate a fractal surface. I have an intuition that this would be true, but I'm not sure how to verify this (other than building models and coming up with a conjecture).