Prometheus Fusion Perfection

Our unit dodecahedron does not lineup with out unit icosahedron:

dodecahedron:
(±1, ±1, ±1)
(0, ±1/φ, ±φ)
(±1/φ, ±φ, 0)
(±φ, 0, ±1/φ)

icosahedron:
(0, ±1, ±φ)
(±1, ±φ, 0)
(±φ, 0, ±1)

We need to scale and rotate the dodecahedron, not sure of the specifics.

I think these are beautiful shapes that may make for good logos (with some framing of course!):