Andreini tessellation

The Andreini tessellations are tilings of three-dimensional space using Platonic and Archimedean solids such that all vertices are identical. There are precisely five such tilings:

• The tiling of cubes
• The tiling of octahedra and cuboctahedra
• The tiling of truncated octahedra[?]
• The tiling of octahedra and tetrahedra
• The tiling of tetrahedra and truncated tetrahedra[?]

The tiling of octahedra and tetrahedra is of special importance since its vertices form a cubic close-packing of spheres. All of these are found in crystal arrangements.

This is not correct. There are 28 such tessellations. See B. Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.