Above is the entire figure, showing all three set of kites. The yellow set below, though, lie along the edges of a rhombic dodecahedron.
The next set, the blue kites, lie along the edges of an octahedron.
Finally, the red set of kites lies along the edges of a cube — the dual to the octahedron delineated by the blue kites.
These images were made using Stella 4d, which is available here.