LiveGraphics3D views of
Jim Lehman's Vector Matrix (LVM)

Mouse to rotate, Shift-mouse to zoom in/out,
s for stereo, s a 2nd time for cross-eyed vs. splay-eyed stereo,
ALT + right mouse button: drag down to remove elements (drag up to add back)

I imagine the green cube as a 2-frequency, volume 24 affair from the RBF concentric hierarchy. For the pentagonal dodecahedron to embrace it in this way, it has to be scaled up from the structural dual to the icosahedron of edges 1 (these two together form the edges of the rhombic triacontahedron -- see link). A scale factor of Sqrt(2) accomplishes this. The blue stellate with the orange ridges is called a concave dodecahedron and space-fills in complement with pentagonal dodecahedra. Jim Lehman uses a repeating pattern of these shapes to form a lattice.

Above we see that the orange segments, known as "rybo keels" on the synergeo list, are actually the long edges of the icosahedron's 3 golden rectangles. These rectangles have short:long dimensions in the ratio of 1 : phi, where phi = (1+sqrt(5))/2.

Above, the orange segments have been removed and we see the six spires (dark blue) which have base triangles congruent to icosahedron faces, and tips at the corners of the 2-frequency cube.

The red octahedron is the volume 4 regular octahedron from the concentric hierarchy. Its edges comprise the long diagonals of the space-filler rhombic dodecahedron faces (not depicted). The icosahedron, in order to inscribe in this octahedron's faces, had to be scaled down from the icosahedron with unit edges (of volume approx 18.51). A scale factor of (sqrt(2)/2))(3-sqrt(5)) is needed.

Above I've added the cuboctahedron (yellow), which has a volume of 20, and is generated when 12 IVM spheres close pack around a nuclear center sphere. This is equivalently the FCC or CCP packing. The red octahedron tips end up in between-sphere voids in this description, but form the anchor points of an alternative IVM (i.e. if those points become spheres, and the current spheres become points, we will still have an IVM).

For Further Reading:

by Kirby Urner using some home-grown Python code
LiveGraphics3D Java applet by Martin Kraus