The Rhombic Dodecahedron can be viewed as a cube with pyramids put on its faces.
These pyramids fit together to fill another cube. Therefore one can tessalate space
as follows: Start with a black and white, checkerboard like, tessalation by cubes.
Subdivide each white cube into the 6 pyramids and add these pyramids to the adjacent
faces of the black cubes. Then one has a tessalation of space by rhombic dodecahedra.
The Rhombic Dodecahedron inside its dual, the cubeoctahedron (wireframe).
Its 4-edged vertices touch the midpoints of the cubeoctahedron faces.
The length of the long diagonal of the rhombic faces equals the distance between the
midpoints of neighboring square faces of the cubeoctahedron.
The endpoints of the short diagonals stay well inside the cubeoctahedron. While it was easy
to construct the rhombic dodecahedron with pyramids on a cube, or also touching the cubeoctahdron
from outside, the image here does not show a construction of the rhombic dodecahedron.